Research Article On a Level-Set Method for Ill-Posed...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 123643 15 pageshttpdxdoiorg1011552013123643

Research ArticleOn a Level-Set Method for Ill-Posed Problems with PiecewiseNonconstant Coefficients

A De Cezaro

Institute of Mathematics Statistics and Physics Federal University of Rio Grande Avenida Italia km 896201-900 Rio Grande RS Brazil

Correspondence should be addressed to A De Cezaro decezaromtmgmailcom

Received 24 July 2012 Revised 16 October 2012 Accepted 16 October 2012

Academic Editor Alicia Cordero

Copyright copy 2013 A De CezaroThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate a level-set-type method for solving ill-posed problems with the assumption that the solutions are piecewise butnot necessarily constant functions with unknown level sets and unknown level values In order to get stable approximate solutionsof the inverse problem we propose a Tikhonov-type regularization approach coupled with a level-set framework We prove theexistence of generalized minimizers for the Tikhonov functional Moreover we prove convergence and stability for regularizedsolutions with respect to the noise level characterizing the level-set approach as a regularization method for inverse problems Wealso show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models

1 Introduction

Since the seminal paper of Santosa [1] level-set techniqueshave been successfully developed and have recently becomea standard technique for solving inverse problems withinterfaces (eg [2ndash10])

In many applications interfaces represent interestingphysical parameters (inhomogeneities heat conductivitybetween materials with different heat capacity and interfacediffusion problems) across which one or more of thesephysical parameters change value in a discontinuousmannerThe interfaces divide the domain Ω sub R119899 in subdomains Ω

119895

with 119895 = 1 119896 of different regions with specific internalparameter profiles Due to the different physical structures ofeach of these regions differentmathematicalmodelsmight bethe most appropriate for describing them Solutions of suchmodels represent a free boundary problem that is one inwhich interfaces are also unknown and must be determinedin addition to the solution of the governing partial differentialequation In general such solutions are determined by aset of data obtained by indirect measurements [2ndash4 11ndash15] Applications include image segmentation problems [12ndash15] optimal shape designer problems [2 16] Stefanrsquos typeproblems [2] inverse potential problems [17ndash19] inverse

conductivityresistivity problems [4 5 10 11 20] amongothers [2ndash4 6 16]

There is often a large variety of priors informationavailable for determining the unknown physical parameterwhose characteristic depends on the given application In thispaper we are interested in inverse problems that consist inthe identification of an unknown quantity 119906 isin 119863(119865) sub 119883that represents all parameter profiles inside the individualsubregions ofΩ from data 119910 isin 119884 where119883 and 119884 are Banachspaces and 119863(119865) will be adequately specified in Section 3 Inthis particular case only the interfaces between the differentregions and possibly the unknown parameter values need tobe reconstructed from the gathered data This process can beformally described by the operator equation

119865 (119906) = 119910 (1)

where 119865 119863(119865) sub 119883 rarr 119884 is the forward operatorNeither existence nor uniqueness of a solution to (1) is

a guarantee For simplicity we assume that for exact data119910 isin 119884 the operator equation (1) admits a solution andwe do not strive to obtain results on uniqueness Howeverin practical applications data are obtained only by indirectmeasurements of the parameter Hence in general exact data

2 Journal of Applied Mathematics

119910 isin 119884 are not known and we have only access to noise data119910120575

isin 119884 whose level of noise 120575 gt 0 is assumed to be known apriori and satisfies

10038171003817100381710038171003817119910120575

minus 11991010038171003817100381710038171003817119884le 120575 (2)

We assume that the inverse problem associated with theoperator equation (1) is ill-posed Indeed it is the case inmany interesting problems [4 6 10 16 20ndash22] Thereforeaccuracy of an approximated solution calls for a regulariza-tion method [21] In this paper we propose a Tikhonov-typeregularization method coupled with a level-set approach toobtain a stable approximation of the unknown level sets andvalues of the piecewise (not necessarily constant) solution of(1)

Many approaches in particular level-set type approacheshave previously been suggested for such problems In [111 23ndash26] level-set approaches for identification of theunknown parameter 119906 with distinct but known piecewiseconstant values were investigated

In [12 17 24] level-set approaches were derived to solveinverse problems assuming that 119906 is defined by severaldistinct constant values In both cases one needs only toidentify the level sets of 119906 that is the inverse problem reducesto a shape identification problem On the other hand whenthe level values of 119906 are also unknown the inverse problembecomes harder since we have to identify both the level setsand the level values of the unknown parameter 119906 In thissituation the dimension of the parameter space increases bythe number of unknown level values Level-set approachesto ill-posed problems with unknown constant level valuesappeared before in [14 16 18 19 27] Level-set regularizationproperties of the approximated solution for inverse problemsare described in [17ndash19 25 28]

However regularization theory for inverse problemswhere the components of the parameter 119906 are variable andhave discontinuities has not been well investigated Indeedlevel-set regularization theory applied to inverse problems[17ndash19] that recover the shape and the values of variablediscontinuous coefficients is unknown to the author Someearly results in the numerical implementation of level-set typemethods were previously used to obtain solutions of ellipticproblems with discontinuous and variable coefficients in [4]

In this paper we propose a level-set type regularizationmethod to ill-posed problems whose solution is composed bypiecewise components which are not necessarily constantsIn other words we introduce a level-set type regularizationmethod to recover the shape and the values of variablediscontinuous coefficients In this framework a level-setfunction is used to parameterize the solution 119906 of (1)We obtain a regularized solution using a Tikhonov-typeregularization method since the level values of 119906 are notconstant and also unknown

In the theoretical point of view the advantage of ourapproach in relation to [2 17ndash19 25 29] is that we areable to obtain regularized solutions to inverse problems with

piecewise solutions that are more general than those coveredby the regularizationmethods proposed beforeWe still proveregularization properties for the approximated solution ofthe inverse problem model (1) where the parameter is anonconstant piecewise solution The topologies needed toguarantee the existence of a minimizer (in a generalizedsense) of the Tikhonov functional (defined in (7)) are quitecomplicated and differ in some key points from [18 19 25]In this particular approach the definition of generalizedminimizers is quite different from other works [17 19 25](see Definition 3) As a consequence the arguments usedto prove the well-posedness of the Tikhonov functionalthe stability and convergence of the regularized solutionsof the inverse problem (1) are quite complicated and needsignificant improvements (see Section 3)

The main applicability advantage of the proposed level-set type method compared to that in the literature is that weare able to apply this method to problems whose solutionsdepend of nonconstant parameters This implies that weare able to handle more general and interesting physicalproblems where the components of the desired parameterare not necessarily homogeneous as those presented beforein the literature [4 6 14 16 18 19 27 30ndash32] Examplesof such interesting physical problems are heat conductionbetween materials of different heat capacity and conductiv-ity interface diffusion processes and many other types ofphysical problems where modeling components are relatedwith embedded boundaries See for example [3 4 6 1930 32] and references therein As a benchmark problem weanalyze two inverse problems modeled by elliptic PDEs withdiscontinuous and variable coefficients

In contrast the nonconstant characteristics of the levelvalues impose different types of theoretical problems sincethe topologies where we are able to provide regularizationproperties of the approximated solution are more compli-cated than the ones presented before [14 16 18 19 27]As a consequence the numerical implementations becomeharder than the other approaches in the literature [18 19 2932]

This paper is outlined as follows in Section 2 weformulate the Tikhonov functional based on the level-setframework In Section 3 we present the general assump-tions needed in this paper and the definition of the setof admissible solutions We prove relevant properties aboutthe admissible set of solutions in particular convergencein suitable topologies We also present relevant propertiesof the penalization functional In Section 4 we prove thatthe proposed method is a regularization method to inverseproblems that is we prove that the minimizers of theproposed Tikhonov functional are stable and convergentwith respect to the noise level in the data In Section 5 asmooth functional is proposed to approximate minimizersof the Tikhonov functional defined in the admissible setof solutions We provide approximation properties and theoptimality condition for the minimizers of the smoothTikhonov functional In Section 6 we present an applicationof the proposed framework to solve some interesting inverseelliptic problems with variable coefficients Conclusions andfuture directions are presented in Section 7

Journal of Applied Mathematics 3

2 The Level-Set Formulation

Our starting point is the assumption that the parameter 119906 in(1) assumes two unknown functional values that is 119906(119909) isin1205951

(119909) 1205952

(119909) ae inΩ sub R119889 whereΩ is a bounded setMorespecifically we assume the existence of amensurable set119863 subsubΩ with 0 lt |119863| lt |Ω| such that 119906(119909) = 1205951(119909) if 119909 isin 119863 and119906(119909) = 120595

2

(119909) if 119909 isin Ω119863 With this framework the inverseproblem that we are interested in in this paper is the stableidentification of both the shape of 119863 and the value function120595119895

(119909) for 119909 belonging to 119863 and to Ω119863 respectively fromobservation of the data 119910120575 isin 119884

We remark that if 1205951(119909) = 1198881 and 1205952(119909) = 1198882 with 1198881and 1198882 unknown constants values the problem of identifying119906 was rigorously studied before in [19] Moreover manyother approaches to this case appear in the literature see[2 19 23 24] and references therein Recently in [18] an 1198712level-set approach to identify the level and constant contrastwas investigated

Our approach differs from the level-set methods pro-posed in [18 19] by considering also the identification ofvariable unknown levels of the parameter 119906 In this situationmany topological difficulties appear in order to have atractable definition of an admissible set of parameters (seeDefinition 3)Generalization to problemswithmore than twolevels is possible applying this approach and following thetechniques derived in [17] As observed before the presentlevel-set approach is a rigorous derivation of a regularizationstrategy for identification of the shape and nonconstant levelsof discontinuous parameters Therefore it can be applied tophysical problems modeled by embedded boundaries whosecomponents are not necessarily piecewise constant [2 17ndash19 25]

In many interesting applications the inverse problemmodeled by (1) is ill-posedTherefore a regularizationmethodmust be applied in order to obtain a stable approximatesolution We propose a regularization method by firstintroducing a parameterization on the parameter space usinga level-set function120601 that belongs to1198671(Ω) Note that we canidentify the distinct level sets of the function 120601 isin 1198671(Ω)withthe definition of the Heaviside projector

119867 1198671

(Ω) 997888rarr 119871infin

(Ω)

120601 997891997888rarr 119867(120601) = 1 if 120601 (119909) gt 00 other else

(3)

Now from the framework introduced above a solution 119906 of(1) can be represented as

119906 (119909) = 1205951

(119909)119867 (120601) + 1205952

(119909) (1 minus 119867 (120601))

= 119875 (120601 1205951

1205952

) (119909)

(4)

With this notation we are able to determine the shapes of 119863as 119909 isin Ω 120601(119909) gt 0 andΩ119863 as 119909 isin Ω 120601(119909) lt 0

The functional level values 1205951(119909) 1205952(119909) are also assumedbe unknown and they should be determined as well

Assumption 1 Weassume that1205951 1205952 isin B = 119891 119891 ismeas-urable and 119891(119909) isin [119898119872] ae in Ω for some constantvalues119898119872

Remark 1 We remark that 119891 isin B implies that 119891 isin 119871infin(Ω)SinceΩ is bounded 119891 isin 1198711(Ω) Moreover

intΩ

119891 (119909) nabla sdot 120593 (119909) 119889119909 le |119872|intΩ

1003816100381610038161003816nabla sdot (120593) (119909)1003816100381610038161003816 119889119909

le |119872|1003817100381710038171003817nabla sdot 120593

10038171003817100381710038171198711(Ω) forall120593 isin 119862

1

0(ΩR

119899

)

(5)

Hence 119891 isin 119861119881(Ω)

Note that in the case that 1205951 and 1205952 assume two distinctconstant values (as covered by the analysis done in [2 18 19]and references therein) the assumptions above are satisfiedHence the level-set approach proposed here generalizes theregularization theory developed in [18 19]

From (4) the inverse problem in (1) with data given as in(2) can be abstractly written as the operator equation

119865 (119875 (120601 1205951

1205952

)) = 119910120575

(6)

Once an approximate solution (120601 1205951

1205952

) of (6) isobtained a corresponding solution of (1) can be computedusing (4)

Therefore to obtain a regularized approximated solutionto (6) we will consider the least square approach combinedwith a regularization term that is minimizing the Tikhonovfunctional

G120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875 (120601 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572

1205731

1003816100381610038161003816119867 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881

(7)

where 1206010and 120595119895

0represent some a priori information about

the true solution 119906lowast of (1)The parameter 120572 gt 0 plays the roleof a regularization parameter and the values of 120573

119894 119894 = 1 2 3

act as scaling factors In other words 120573119894 119894 = 1 2 3 needs to

be chosen a priori but independent of the noise level 120575 Inpractical 120573

119894 119894 = 1 2 3 can be chosen in order to represent

a priori knowledge of features of the parameter solution 119906andor to improve the numerical algorithmAmore completediscussion about how to choose 120573

119894 119894 = 1 2 3 is provided in

[17ndash19]The regularization strategy in this context is based on

119879119881-1198671-119879119881 penalization The term on 1198671-norm acts simul-taneously as a control on the size of the norm of the level-set function and a regularization on the space 1198671 The termon 119861119881 is a variational measure of119867(120601) It is well known that


the 119861119881 seminorm acts as a penalizing for the length of theHausdorff measure of the boundary of the set 119909 120601(119909) gt 0(see [33 Chapter 5] for details) Finally the last term on 119861119881is a variational measure of 120595119895 that acts as a regularizationterm on the setB This Tikhonov functional extends the onesproposed in [16 17 19 23 24] (based on119879119881-1198671 penalization)

Existence of minimizers for the functional (7) in the1198671timesB2 topology does not follow by direct arguments since theoperator 119875 is not necessarily continuous in this topologyIndeed if1205951 = 1205952 = 120595 is a continuous function at the contactregion then 119875(1206011 1205952 120595) = 120595 is continuous and the standardTikhonov regularization theory to the inverse problem holdstrue [21] On the other hand in the interesting case where 1205951and 1205952 represent the level of discontinuities of the parameter119906 the analysis becomes more complicated and we need adefinition of generalized minimizers (see Definition 3) inorder to handle these difficulties

3 Generalized Minimizers

As already observed in [25] if 119863 sub Ω with H119899minus1(120597119863) lt infinwhere H119899minus1(119878) denotes the (119899 minus 1)-dimensional Hausdorff-measure of the set 119878 then the Heaviside operator 119867 maps1198671

(Ω) into the set

V = 120594119863 119863 sub Ω measurable H119899minus1 (120597119863) lt infin (8)

Therefore the operator 119875 in (4) maps 1198671(Ω) times B2 into theadmissible parameter set

119863 (119865) = 119906 = 119902 (119907 1205951

1205952

) 119907 isinV 1205951

1205952

isin B (9)

where

119902 V times B2

ni (119907 1205951

1205952

) 997891997888rarr 1205951

119907 + 1205952

(1 minus 119907) isin 119861119881 (Ω)

(10)

Consider the model problem described in Section 1 Inthis paper we assume the following

(A1) Ω sube R119899 is bounded with piecewise 1198621 boundary 120597Ω(A2) The operator 119865 119863(119865) sub 1198711(Ω) rarr 119884 is continuous

on119863(119865) with respect to the 1198711(Ω) topology(A3) 120576 120572 and 120573

119895 119895 = 1 2 3 denote positive parameters

(A4) Equation (1) has a solution that is there exists 119906lowastisin

119863(119865) satisfying119865(119906lowast) = 119910 and a function 120601

lowastisin 1198671

(Ω)

satisfying |nabla120601lowast| = 0 in the neighborhood of 120601

lowast= 0

such that 119867(120601lowast) = 119911

lowast for some 119911

lowastisin V Moreover

there exist functional values 1205951lowast 1205952

lowastisin B such that

119902(119911lowast 1205951

lowast 1205952

lowast) = 119906lowast

For each 120576 gt 0 we define a smooth approximation to theoperator 119875 by

119875120576(120601 1205951

1205952

) = 1205951

119867120576(120601) + 120595

2

(1 minus 119867120576(120601)) (11)

where119867120576is the smooth approximation to119867 described by

119867120576(119905) =

1 +119905

120576for 119905 isin [minus120576 0]

119867 (119905) for 119905 isin R

[minus120576 0]

(12)

Remark 2 It is worth noting that for any 120601119896isin 1198671

(Ω)119867120576(120601119896)

belongs to 119871infin(Ω) and satisfies 0 le 119867120576(120601119896) le 1 ae in Ω for

all 120576 gt 0 Moreover taking into account that120595119895 isin B it followsthat the operators 119902 and 119875

120576 as above are well defined

In order to guarantee the existence of a minimizer of G120572

defined in (7) in the space1198671(Ω) timesB2 we need to introducea suitable topology such that the functional G

120572has a closed

graphic Therefore the concept of generalized minimizers(compare with [17 25]) in this paper is as follows

Definition 3 Let the operators 119867 119875 119867120576 and 119875

120576be defined

as above and the positive parameters 120572 120573119895 and 120576 satisfy the

Assumption (A3)A quadruple (119911 120601 1205951 1205952) isin 119871infin(Ω) times 1198671(Ω) times 119861119881(Ω)2

is called admissible when

(a) there exists a sequence 120601119896 of 1198671(Ω) functions

satisfying lim119896rarrinfin

120601119896minus 1206011198712(Ω)= 0

(b) there exists a sequence 120576119896 isin R+ converging to zero

such that lim119896rarrinfin

119867120576119896

(120601119896) minus 119911

1198711(Ω)= 0

(c) there exist sequences 1205951119896119896isinN and 1205952

119896119896isinN belonging

to 119861119881 cap 119862infin(Ω) such that10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (13)

(d) a generalized minimizer of G120572is considered to be

any admissible quadruple (119911 120601 1205951 1205952)minimizing

G120572(119911 120601 120595

1

1205952

) =10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

)

(14)

on the set of admissible quadruples Here the func-tional 119877 is defined by

119877 (119911 120601 1205951

1205952

) = 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881 (15)

and the functional 120588 is defined as

120588 (119911 120601) = inf lim inf119896rarrinfin

[1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)]

(16)

The infimum in (16) is taken over all sequences 120576119896 and 120601

119896

characterizing (119911 120601 1205951 1205952) as an admissible quadruple

The convergence |120595119895119896|119861119881

rarr |120595119895

|119861119881

in item (c) inDefinition 3 is in the sense of variation measure [33 Chapter5] The incorporation of item (c) in the Definition 3 impliesthe existence of the Γ-limit of sequences of admissiblequadruples [25 34]This appears in the proof of Lemmas 7 8and 11 where we prove that the set of admissible quadruplesis closed in the defined topology (see Lemmas 7 and 8) andin the weak lower semicontinuity of the regularization func-tional 119877 (see Lemma 11) The identification of nonconstant


level values 120595119895 implies in a different definition of admissiblequadruples

As a consequence the arguments in the proof of regular-ization properties of the level-set approach are the principaltheoretical novelty and the difference between our definitionof admissible quadruples and the ones in [18 19 25]

Remark 4 For 119895 = 1 2 let 120595119895 isin B cap 119862infin(Ω) 120601 isin 1198671(Ω) besuch that |nabla120601| = 0 in the neighborhood of the level-set 120601(119909) =0 and 119867(120601) = 119911 isin V For each 119896 isin N set 120595119895

119896= 120595119895 and

120601119896= 120601 Then for all sequences of 120576

119896119896isinN of positive numbers

converging to zero we have10038171003817100381710038171003817119867120576119896

(120601119896) minus 119911

100381710038171003817100381710038171198711(Ω)=10038171003817100381710038171003817119867120576119896

(120601119896) minus 119867 (120601)

100381710038171003817100381710038171198711(Ω)

= int(120601)minus1[minus1205761198960]

10038161003816100381610038161003816100381610038161003816

1 minus120601

120576119896

10038161003816100381610038161003816100381610038161003816

119889119909

le int

0

minus120576119896

int(120601)minus1(120591)

1119889120591

le meas (120601)minus1 (120591) int0

minus120576119896

1119889119905 997888rarr 0

(17)

Here we use the fact that |nabla120601| = 0 in the neighborhood of120601 = 0 implies that 120601 is a local diffeomorphism togetherwith a coarea formula [33 Chapter 4] Moreover 120595119895

119896119896isinN in

B cap 119862infin(Ω) satisfies Definition 3 item (c)Hence (119911 120601 1205951 1205952) is an admissible quadruple In partic-

ular we conclude from the general assumption above that theset of admissible quadruple satisfying 119865(119906) = 119910 is not empty

31 Relevant Properties of Admissible Quadruples Our firstresult is the proof of the continuity properties of operators 119875

120576

119867120576 and 119902 in suitable topologies Such result will be necessary

in the subsequent analysisWe start with an auxiliary lemma that is well known (see

eg [35]) We present it here for the sake of completeness

Lemma 5 Let Ω be a measurable subset of R119899 with finitemeasure

If (119891119896) isin B is a convergent sequence in 119871119901(Ω) for some 119901

1 le 119901 lt infin then it is a convergent sequence in 119871119901(Ω) for all1 le 119901 lt infin

In particular Lemma 5 holds for the sequence 119911119896=

119867120576(120601119896)

Proof See [35 Lemma 21]

The next two lemmas are auxiliary results in order tounderstand the definition of the set of admissible quadruples

Lemma 6 Let Ω be as in assumption (A1) and 119895 = 1 2

(i) Let 119911119896119896isinN be a sequence in 119871infin(Ω) with 119911

119896isin [119898119872]

ae converging in the 1198711(Ω)-norm to some element 119911and 120595119895

119896119896isinN a sequence in B converging in the 119861119881-

norm to some 120595119895 isin B Then 119902(119911119896 1205951

119896 1205952

119896) converges to

119902(119911 1205951

1205952

) in 1198711(Ω)

(ii) Let (119911 120601) isin 1198711(Ω) times 1198671(Ω) be such that 119867120576(120601) rarr

119911 in 1198711(Ω) as 120576 rarr 0 and let 1205951 1205952 isin B Then119875120576(120601 1205951

1205952

) rarr 119902(119911 1205951

1205952

) in 1198711(Ω) as 120576 rarr 0(iii) Given 120576 gt 0 let 120601

119896119896isinN be a sequence in 1198671(Ω) con-

verging to 120601 isin 1198671(Ω) in the 1198712-normThen119867120576(120601119896) rarr

119867120576(120601) in 1198711(Ω) as 119896 rarr infin Moreover if 120595119895

119896119896isinN

are sequences in B converging to some 120595119895 in B withrespect to the 1198711(Ω)-norm then 119902(119867

120576(120601119896) 1205951

119896 1205952

119896) rarr

119902(119867120576(120601) 120595

1

1205952

) in 1198711(Ω) as 119896 rarr infin

Proof Since Ω is assumed to be bounded we have 119871infin(Ω) sub1198711

(Ω) and 119861119881(Ω) is continuous embedding in 1198712(Ω) [33] Toprove (i) notice that10038171003817100381710038171003817119902 (119911119896 1205951

119896 1205952

119896) minus 119902 (119911 120595

1

1205952

)100381710038171003817100381710038171198711(Ω)

=100381710038171003817100381710038171205951

119896119911119896+ 1205952

119896(1 minus 119911

119896) minus 1205951

119911 minus 1205952

(1 minus 119911)100381710038171003817100381710038171198711(Ω)

le10038171003817100381710038171199111198961003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205951

119896minus 1205951100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205951100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

+10038171003817100381710038171 minus 119911119896

1003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205952

119896minus 1205952100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205952100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

119896rarrinfin

997888rarr 0

(18)

Here we use Lemma 5 in order to guarantee the convergenceof 119911119896to 119911 in 1198712(Ω)

Assertion (ii) follows with similar arguments and the factthat119867

120576(120601) isin 119871

infin

(Ω) for all 120576 gt 0As 119867

120576(120601119896) minus 119867

120576(120601)1198711(Ω)le 120576minus1radicmeas(Ω)120601

119896minus 1206011198712(Ω)

the first part of assertion (iii) follows The second part of theassertion (iii) holds by a combination of the inequality aboveand inequalities in the proof of assertion (i)

Lemma 7 Let 120595119895119896119896isinN be a sequence of functions satisfying

Definition 3 converging in 1198711(Ω) to some 120595119895 for 119895 = 1 2 Then120595119895 also satisfies Definition 3

Proof (sketch of the proof) Let 119896 isin N and 119895 = 1 2 Since 120595119895119896

satisfies Definition 3 120595119895119896isin 119861119881 From [33 Theorem 2 p 172]

there exist sequences 120595119895119896119897119897isinN in 119861119881 times 119862infin(Ω) such that

120595119895

119896119897

119897rarrinfin

997888rarr 120595119895

119896in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(19)

In particular for the subsequence 120595119895119896119897(119896)

119896isinN it follows that

120595119895

119896119897(119896)

119896rarrinfin

997888rarr 120595119895 in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897(119896)

10038161003816100381610038161003816119861119881

119896rarrinfin

997888rarr1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881

(20)

Moreover by assumption 120595119895 isin 1198711

(Ω) From the lowersemicontinuity of variational measure (see [33 Theorem 1 p


172]) (20) and the definition of 119861119881 space it follows that120595119895

isin 119861119881

In the following lemmawe prove that the set of admissiblequadruples is closed with respect to the 1198711(Ω) times 1198712(Ω) times(1198711

(Ω))2 topology

Lemma 8 Let (119911119896 120601119896 1205951

119896 1205952

119896) be a sequence of admissible

quadruples converging in 1198711(Ω) times 1198712(Ω) times (1198711(Ω))2 to some(119911 120601 120595

1

1205952

) with 120601 isin 1198671(Ω) Then (119911 120601 1205951 1205952) is also anadmissible quadruple

Proof (sketch of the proof) Let 119896 isin N Since (1199111119896 1206011

119896 1205951

119896 1205952

119896)

is an admissible quadruple it follows from Definition 3 thatthere exist sequences 120601

119896119897119897isinN in 119867

1

(Ω) 1205951119896119897119897isinN 120595

2

119896119897119897isinN

in 119861119881 times 119862infin

(Ω) and a correspondent sequence 120576119897119896119897isinN

converging to zero such that

120601119896119897

119897rarrinfin

997888rarr 120601119896

in 1198712 (Ω)

119867120576119897

119896

(120601119896119897)119897rarrinfin

997888rarr 119911119896

in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(21)

Define the monotone increasing function 120591 N rarr N

such that for every 119896 isin N it holds

120576120591(119896)

119896le1

2120576120591(119896minus1)

119896minus1

1003817100381710038171003817120601119896120591(119896) minus 12060111989610038171003817100381710038171198712(Ω)

le1

119896

100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)le1

119896

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(22)

Hence for each 119896 isin N1003817100381710038171003817120601 minus 120601119896120591(119896)

10038171003817100381710038171198712(Ω)

le1003817100381710038171003817120601 minus 120601119896

10038171003817100381710038171198712(Ω)+1003817100381710038171003817120601119896120591(119896) minus 120601119896

10038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

le1003817100381710038171003817119911 minus 119911119896

10038171003817100381710038171198711(Ω)+100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)

(23)

From (22)

lim119896rarrinfin

1003817100381710038171003817120601 minus 120601119896120591(119896)10038171003817100381710038171198712(Ω)

= 0

lim119896rarrinfin

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

= 0

(24)

Moreover with the same arguments as Lemma 7 it followsthat

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (25)

and 120595119895 isin 119861119881(Ω) Therefore it remains to prove that (119911 120601 12059511205952

) is an admissible quadruple From Definition 3 and

Lemma 7 it is enough to prove that 119911 isin 119871infin(Ω) If this isnot the case there would exist a Ω1015840 sub Ω with |Ω1015840| gt 0 and120574 gt 0 such that 119911(119909) gt 1 + 120574 in Ω1015840 (the other case 119911(119909) lt minus120574is analogous) Since (119867

120576120591(119896)

119896

(120601119896120591(119896)

))(119909) isin [0 1] ae in Ω for119896 isin N (see remark after Definition 3) we would have

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

ge100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω1015840)

ge 12057410038161003816100381610038161003816Ω101584010038161003816100381610038161003816 119896 isin N

(26)

contradicting the second limit in (24)

32 Relevant Properties of the Penalization Functional In thefollowing lemmas we verify properties of the functional 119877which are fundamental for the convergence analysis outlinedin Section 4 In particular these properties imply that thelevel sets of G

120572are compact in the set of admissible quadru-

ple that is G120572assumes a minimizer on this set First we

prove a lemma that simplifies the functional 119877 in (15) Herewe present the sketch of the proof For more details see thearguments in [19 Lemma 3]

Lemma9 Let (119911 120601 1205951 1205952) be an admissible quadrupleThenthere exist sequences 120576

119896119896isinN 120601119896119896isinN and 120595

119895

119896119896isinN as in the

Definition 3 such that

119877 (119911 120601 1205951

1205952

) = lim119896rarrinfin

1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(27)Proof (sketch of the proof) For each 119897 isin N the definition of119877 (see Definition 3) guaranties the existence of sequences 120576119897

119896

120601119895

119896119897 isin 119867

1

(Ω) and 120595119895119896119897 isin B such that

119877 (119911 120601 1205951

1205952

)

= lim119897rarrinfin

lim inf119896rarrinfin

1205731

100381610038161003816100381610038161003816119867120576119897

119896

(120601119896119897)100381610038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896119897 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896119897minus 120595119895

0

10038161003816100381610038161003816119861119881

(28)Now a similar extraction of subsequences as in Lemma 8complete the proof

In the following we prove two lemmas that are essentialto the proof of well posedness of the Tikhonov functional (7)

Lemma 10 The functional 119877 in (15) is coercive on the set ofadmissible quadruples In other words given any admissiblequadruple (119911 120601 1205951 1205952) one has


119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)

Proof (sketch of the proof) Let (119911 120601 1205951 1205952) be an admissiblequadruple From [17 Lemma 4] it follows that

120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)

Now from (30) and the definition of 119877 in (15) we have

(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)

concluding the proof

Lemma 11 The functional 119877 in (15) is weak lower semicon-tinuous on the set of admissible quadruples that is given asequence (119911

119896 120601119896 1205951

119896 1205952

119896) of admissible quadruples such that

119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)

for some admissible quadruple (119911 120601 1205951 1205952) then

119877 (119911 120601 1205951

1205952

) le lim inf119896isinN

119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)

Proof The functional 120588(119911 120601) is weak lower semicontinuous(cf [17 Lemma 5]) As120595119895

119896isin 119861119881 it follows from [33Theorem

2 p 172] that there exist sequences 120595119895119896119897 isin 119861119881cap119862

infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)

le 1119897 From a diagonal argument we canextract a subsequence 120595119895

119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr

120595119895 in 1198711(Ω) as 119896 rarr infin Let 120585 isin 1198621

119888(ΩR119899) |120585| le 1 Then

from [33 Theorem 1 p 167] it follows that

intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]

le lim119896rarrinfin

[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)

1003817100381710038171003817nabla sdot 1205851003817100381710038171003817119871infin(Ω) |

Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]

le lim inf119896rarrinfin

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)

Thus form the definition of | sdot |119861119881

(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)

Now the lemma follows from the fact that the functional119877 in (15) is a linear combination of lower semicontinuousfunctionals

4 Convergence Analysis

In the following we consider any positive parameter120572 120573119895 119895 = 1 2 3 as in the general assumption to this paper

First we prove that the functionalG120572in (14) is well posed

Theorem 12 (well-posedness) The functional G120572in (14)

attains minimizers on the set of admissible quadruples

Proof Notice that the set of admissible quadruples is notempty since (0 0 0 0) is admissible Let (119911

119896 120601119896 1205951

119896 1205952

119896) be a

minimizing sequence forG120572 that is a sequence of admissible

quadruples satisfying G120572(119911119896 120601119896 1205951

119896 1205952

119896) rarr inf G

120572le

G120572(0 0 0 0) lt infin Then G

120572(119911119896 120601119896 1205951

119896 1205952

119896) is a bounded

sequence of real numbers Therefore (119911119896 120601119896 1205951

119896 1205952

119896) is uni-

formly bounded in119861119881times1198671(Ω)times1198611198812Thus from the SobolevEmbedding Theorem [33 36] we guarantee the existence ofa subsequence (denoted again by (119911

119896 120601119896 1205951

119896 1205952

119896)) and the

existence of (119911 120601 1205951 1205952) isin 1198711(Ω) times 1198671(Ω) times 1198611198812 such that120601119896rarr 120601 in 1198712(Ω) 120601

119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1

(Ω) Moreover 119911 1205951 and 1205952 isin 119861119881 See [33Theorem 4 p 176]

From Lemma 8 we conclude that (119911 120601 1205951 1205952) is anadmissible quadruple Moreover from the weak lower semi-continuity of 119877 (Lemma 11) together with the continuity of 119902(Lemma 6) and continuity of 119865 (see the general assumption)we obtain

inf G120572= lim119896rarrinfin

G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)

proving that (119911 120601 1205951 1205952)minimizesG120572

In what follows we will denote a minimizer of G120572by

(119911120572 120601120572 1205951

120572 1205952

120572) In particular the functional G

120572in (50) attains

a generalized minimizer in the sense of Definition 3 In thefollowing theorem we summarize some convergence resultsfor the regularizedminimizersThese results are based on theexistence of a generalizedminimum norm solutions


Definition 13 An admissible quadruple (119911dagger 120601dagger 1205951dagger 1205952dagger) iscalled an 119877-minimizing solution if it satisfies

(i) 119865(119902(119911dagger 1205951dagger 1205952dagger)) = 119910(ii) 119877(119911dagger 120601dagger 1205951dagger 1205952dagger) = 119898119904 = inf119877(119911 120601 1205951 1205952) (119911120601 1205951

1205952

) is an admissible quadruple and 119865(119902(1199111205951

1205952

)) = 119910

Theorem 14 (119877-minimizing solutions) Under the generalassumptions of this paper there exists a119877-minimizing solution

Proof From the general assumption on this paper andRemark 4 we conclude that the set of admissible quadruplesatisfying 119865(119902(119911 1205951 1205952)) = 119910 is not empty Thus 119898119904 in (ii)is finite and there exists a sequence (119911

119896 120601119896 1205951

119896 1205952

119896)119896isinN of

admissible quadruple satisfying

119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)

Now form the definition of 119877 it follows that the sequences120601119896119896isinN 119911119896119896isinN and 120595

119895

119896119895=12

119896isinNare uniformly bounded in

1198671

(Ω) and 119861119881(Ω) respectively Then from the SobolevCompact Embedding Theorem [33 36] we have (up tosubsequences) that

120601119896997888rarr 120601

dagger in 1198712 (Ω)

119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)

Lemma 8 implies that (119911dagger 120601dagger 1205951dagger 1205952dagger) is an admissiblequadruple Since 119877 is weakly lower semicontinuous (cfLemma 11) it follows that

119898119904 = lim inf119896rarrinfin

119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)

Moreover we conclude from Lemma 6 that

119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger

)) = lim119896rarrinfin

119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)

Thus (119911dagger 120601dagger 1205951dagger 1205952dagger) is an 119877-minimizing solution

Using classical techniques from the analysis of Tikhonovregularization methods (see [21 37]) we present in thefollowing themain convergence and stability theorems of thispaper The arguments in the proof are somewhat different ofthose presented in [18 19] But for sake of completeness wepresent the proof

Theorem 15 (convergence for exact data) Assume that onehas exact data that is 119910120575 = 119910 For every 120572 gt 0 let (119911

120572 120601120572

1205951

120572 1205952

120572) denote a minimizer of G

120572on the set of admissible

quadruples Then for every sequence of positive numbers120572119896119896isinN converging to zero there exists a subsequence denoted

again by 120572119896119897isinN such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896

) is stronglyconvergent in 1198711(Ω) times 1198712(Ω) times (1198711(Ω))2 Moreover the limit isa solution of (1)

Proof Let (119911dagger 120601dagger 1205951dagger 1205952dagger) be an 119877-minimizing solution of(1)mdashits existence is guaranteed byTheorem 14 Let 120572

119896119896isinN be

a sequence of positive numbers converging to zero For each119896 isin N denote (119911

119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896

) to be aminimizer of 119866

120572119896

Then for each 119896 isin N we have

119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) it follows from

(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)

Moreover from the assumption on the sequence 120572119896 it

follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)

From (41) and Lemma 10 we conclude that sequences 120601119896

119911119896 and 120595119895

119896 are bounded in 1198671(Ω) and 119861119881 respectively

for 119895 = 1 2 Using an argument of extraction of diagonalsubsequences (see proof of Lemma 8) we can guarantee theexistence of an admissible quadruple ( 120601 1 2) such that

(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2

) in 1198711 (Ω) times 1198712 (Ω) times (1198711 (Ω))2

(43)

Now from Lemma 6(i) it follows that 119902( 1 2) =

lim119896rarrinfin

119902(119911119896 1205951

119896 1205952

119896) in 1198711(Ω) Using the continuity of the

operator 119865 together with (40) and (42) we conclude that

119910 = lim119896rarrinfin

119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)

On the other hand from the lower semicontinuity of 119877 and(41) it follows that

119877 ( 120601 1

2

) le lim inf119896rarrinfin

119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le lim sup119896rarrinfin

119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)



Theorem 16 (stability) Let 120572 = 120572(120575) be a function satisfyinglim120575rarr0

120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1

= 0 Moreover let120575119896119896isinN be a sequence of positive numbers converging to zero

and 119910120575119896 isin 119884 corresponding noisy data satisfying (2) Thenthere exists a subsequence denoted again by 120575

119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896

) convergesin 1198711(Ω) times 1198712(Ω) times (1198711(Ω))2 to solution of (1)

Proof Let (119911dagger 120601dagger 1205951dagger 1205951dagger) be an 119877-minimizer solution of(1) (such existence is guaranteed by Theorem 14) For each119896 isin N let (119911

119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be

a minimizer of 119866120572(120575119896) Then for each 119896 isin N we have

119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)

From (46) and the definition of 119866120572119896

it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)

Taking the limit as 119896 rarr infin in (47) it follows fromtheorem assumptions on 120572(120575

119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0

lim sup119896rarrinfin

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)

With the same arguments as in the proof of Theorem 15 weconclude that at least a subsequence that we denote again by(119911119896 120601119896 1205951

119896 1205952

119896) converges in 1198711(Ω)times1198712(Ω)times(1198711(Ω))2 to some

admissible quadruple (119911 120601 1205951 1205952) Moreover by taking thelimit as 119896 rarr infin in (46) it follows from the assumption on 119865and Lemma 6 that

119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)

The functional G120572defined in (14) is not easy to be

handled numerically that is we are not able to derive asuitable optimality condition to the minimizers ofG

120572 In the

following section we work in sight to surpass such difficulty

5 Numerical Solution

In this section we introduce a functional which can behandled numerically and whose minimizers are ldquonearrdquo to the

minimizers ofG120572 LetG

120576120572be the functional defined by

G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952

) is defined in (11) ThefunctionalG

120576120572is well posed as the following lemma shows

Lemma 17 Given positive constants 120572 120576 120573119895as in the general

assumption of this paper 1206010isin 1198671

(Ω) and 1205951198950isin B 119895 =

1 2 Then the functional G120576120572

in (50) attains a minimizer on1198671

(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2

le G120576120572(0 0 0) lt infin there exists a minimizing

sequence (120601119896 1205951

119896 1205952

119896) in1198671(Ω) times B2 satisfying

lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)

Then for fixed 120572 gt 0 the definition of G120576120572

in (50) impliesthat the sequences 120601

119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)

and (119861119881)2 respectively Therefore from Banach-Alaoglu-Bourbaki Theorem [38] 120601

119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595

119895 in 1198711(Ω) 119895 = 1 2 Now asimilar argument as in Lemma 7 implies that 120595119895 isin B for119895 = 1 2 Moreover by the weak lower semicontinuity of the1198671-norm [38] and |sdot|

119861119881measure (see [33Theorem 1 p 172])

it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671 le lim inf119896rarrinfin

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881le lim inf119896rarrinfin

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)

The compact embedding of 1198671(Ω) into 1198712(Ω) [36]implies in the existence of a subsequence of 120601

119896 (that we

denote with the same index) such that 120601119896rarr 120601 in 1198712(Ω)

It follows from Lemma 6 and [33 Theorem 1 p 172] that|119867120576(120601)|119861119881le lim inf

119896rarrinfin|119867120576(120601119896)|119861119881 Hence from continuity


of 119865 in 1198711 continuity of 119902 (see Lemma 6) together with theestimates above we conclude that

G120576120572(120601 1205951

1205952

) le lim119896rarrinfin

10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572(1205731lim inf119896rarrinfin

1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881

+ 1205732lim inf119896rarrinfin

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)

Therefore (120601 1205951 1205952) is a minimizer ofG120576120572

In the sequel we prove that when 120576 rarr 0 theminimizersofG120576120572

approximate aminimizer of the functionalG120572 Hence

numerically the minimizer of G120576120572

can be used as a suitableapproximation for the minimizers ofG

120572

Theorem 18 Let 120572 and 120573119895be given as in the general assump-

tion of this paper For each 120576 gt 0 denote by (120601120576120572 1205951

120576120572 1205952

120576120572)

a minimizer of G120576120572

(that exists form Lemma 17) Then thereexists a sequence of positive numbers 120576

119896rarr 0 such that

(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952

120576119896120572) converges strongly in 1198711(Ω) times

1198712

(Ω) times (1198711

(Ω))2 and the limit minimizes G

120572on the set of

admissible quadruples

Proof Let (119911120572 120601120572 1205951

120572 1205952

120572) be a minimizer of the functional

G120572on the set of admissible quadruples (cf Theorem 12)

From Definition 3 there exists a sequence 120576119896 of positive

numbers converging to zero and corresponding sequences120601119896 in 1198671(Ω) satisfying 120601

119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572

in 1198711(Ω) and finally sequences 120595119895119896 in 119861119881 times 119862infin

119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895

|119861119881 Moreover we can further assume (see

Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896

) be a minimizer of G120576119896120572 Hence (120601

120576119896

1205951

120576119896

1205952

120576119896

) belongs to 1198671(Ω) times B2 (see Lemma 17) The sequences119867120576119896

(120601120576119896

) 120601120576119896

and 120595119895120576119896

are uniformly bounded in 119861119881(Ω)1198671

(Ω) and 119861119881(Ω) for 119895 = 1 2 respectively Form compactembedding (see Theorems [36] and [33 Theorem 4 p 176])there exist convergent subsequenceswhose limits are denoted

by 120601 and 119895 belonging to 119861119881(Ω)1198671(Ω) and 119861119881(Ω) for119895 = 1 2 respectively

Summarizing we have 120601120576119896

rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671

(Ω) times 119871(Ω) is an admissible quadruple (cfLemma 8)

From the definition of 119877 Lemma 6 and the continuity of119865 it follows that

10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572 lim sup119896rarrinfin

(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952

120572) as a minimizer ofG

120572

51 Optimality Conditions for the Stabilized Functional Fornumerical purposes it is convenient to derive first-orderoptimality conditions for minimizers of functionalG

120572 Since

119875 is a discontinuous operator it is not possible Howeverthanks to Theorem 16 the minimizers of the stabilizedfunctionals G

120576120572can be used for approximate minimizers of

the functional G120572 Therefore we consider G

120576120572in (50) with


119884 a Hilbert space and we look for the Gateaux directionalderivatives with respect to 120601 and the unknown120595119895 for 119895 = 1 2

Since 1198671015840

120576(120601) is self-adjoint (note that 1198671015840

120576(119905) =

1120576 119905 isin (minus1205760)

0 other else) we can write the optimality conditions for thefunctionalG

120576120572in the form of the system

120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)

Here 120584(119909) represents the external unit normal quadruple at119909 isin 120597Ω and

119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)

It is worth noticing that the derivation of (57) (58)and (59) is purely formal since the 119861119881 seminorm is notdifferentiable Moreover the terms |nabla119867

120576(120601)| and |nabla(120595119895 minus 120595119895

0)|

appear in the denominators of (57) (58) (59) (60) (61) and(62) respectively

In Section 6 systems (57) (58) (59) (60) (61) and (62)are used as starting point for the derivation of a level-set-typemethod

6 Inverse Elliptic Problems

In this section we discuss the proposed level-set approachand its application in some physical problems modeledby elliptic PDEs We also discuss briefly the numericalimplementations of the iterative method based on the level-set approach We remark that in the case of noise data theiterative algorithm derived by the level-set approach needs anearly stooping criteria [21]

61 The Inverse Potential Problem In this subsection weapply the level-set regularization framework in an inversepotential problem [18 22 32] Differently from [10 18 19 2529 32 39] we assume that the source 119906 is not necessarilypiecewise constant For relevant applications of the inversepotential problem see [20 22 32 39] and references therein

The forward problem consists of solving the Poissonboundary value problem

minusnabla sdot (120590nabla119908) = 119906 in Ω

1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)

on a given domain Ω sub R119899 with 120597Ω Lipschitz for a givensource function 119906 isin 1198712(Ω) and a boundary function 119892 isin1198712

(120597Ω) In (71) 120584 represents the outer normal vector to 120597Ωand 120590 is a known sufficient smooth function Note thatdepending of 120574

1 1205742isin 0 1 we have Dirichlet Neumann or

Robin boundary condition In this paper we only considerthe case of Dirichlet boundary condition that correspondsto 1205741= 1 and 120574

2= 0 in (71) Therefore it is well known

that there exists a unique solution 119908 isin 1198671(Ω) of (71) with119908 minus 119892 isin 119867

1

0(Ω) [40]

Assuming homogeneousDirichlet boundary condition in(63) the problem can be modeled by the operator equation

1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)

The corresponding inverse problem consists in recov-ering the 1198712 source function 119906 from measurements ofthe Cauchy data of its corresponding potential 119908 on theboundary ofΩ

Using this notation the inverse potential problem canbe written in the abbreviated form 119865

1(119906) = 119910

120575 where theavailable noisy data 119910120575 isin 119871

2

(120597Ω) has the same meaningas in (2) It is worth noticing that this inverse problem hasin general nonunique solution [22] Therefore we restrictour attention to minimum-norm solutions [21] Sufficientconditions for identifiability are given in [20] Moreoverwe restrict our attention to solve the inverse problem (64)in 119863(119865) that is we assume the unknown parameter 119906 isin

119863(119865) as defined in Section 3 Note that in this situation theoperator 119865

1is linear However the inverse potential problem

is well known to be exponentially ill-posed [20] Thereforethe solution calls for a regularization strategy [20ndash22]

The following lemma implies that the operator 1198651satisfies

the assumption (A2)

Lemma 19 The operator 1198651 119863(119865) sub 119871

1

(Ω) rarr 1198712

(120597Ω) iscontinuous with respect to the 1198711(Ω) topology

Proof It is well known from the elliptic regularity theory [40]that 119908

1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080

the respective solution of (63) Then from the linearity and


continuity of the trace operator from1198671(Ω) to 1198712(120597Ω) [40]we have that

10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)

where we use Lemma 5 to obtain the last inequality There-fore 119865

1is sequentially continuous on the 1198711(Ω) topology

Since 1198711(Ω) is a metrizable spaces [38] the proof is com-plete

611 A Level-Set Algorithm for the Inverse Potential ProblemWe propose an explicit iterative algorithm derived from theoptimality conditions (57) (58) (59) (60) (61) and (62) forthe Tikhonov functionalG

120576120572

For the inverse potential problem with Dirichlet bound-ary condition (120574

1= 1 and 120574

2= 0) the algorithm reads as

shown in Algorithm 20

Algorithm 20 (iterative algorithm based on the level-setapproach for the inverse potential problem) Given 120590 and 119892

(1) evaluate the residual 119903119896= 1198651(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves

minusnabla sdot (120590nabla119908119896) = 119875120576(120601119896 1205951

119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)

(4) update the level-set function 120601119896and the level values

120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)

Each step of this iterative method consists of three parts(see Algorithm 20) (1) the residual 119903

119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895

119896) is evaluated (this requires solving one elliptic BVP of

Dirichlet type) (2) the 1198712-solution ℎ119896of the adjoint problem

for the residual is evaluated (this corresponds to solving oneelliptic BVP of Dirichlet type) (3) the update 120575120601

119896for the

level-set function and the updates 120575120595119895119896for the level values are

evaluated (this corresponds to multiplying two functions)

In [29] a level-set method was proposed where theiteration is based on an inexact Newton type method Theinner iteration is implemented using the conjugate gradientmethod Moreover the regularization parameter 120572 gt 0 iskept fixed In contrast to [29] in Algorithm 20 we define120575119905 = 1120572 (as a time increment) in order to derive anevolution equation for the level-set function Therefore weare looking for a fixed-point equation related to the system ofoptimality conditions for the Tikhonov functionalMoreoverthe iteration is based on a gradient-type method as in [18]

62 The Inverse Problem in Nonlinear ElectromagnetismMany interesting physical problems are model by quasilin-ear elliptic equations Examples of applications include theidentification of inhomogeneity inside nonlinear magneticmaterials form indirect or local measurements Electromag-netic nondestructive tests aim to localize cracks or inhomo-geneities in the steel production process where impuritiescan be described by a piecewise smooth function [2ndash4 11]

In this section we assume that119863 subsub Ω is measurable and

119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)

where 1205951 1205952isin B and119898 gt 0



119908 = 119892 on 120597Ω(70)

whereΩ sub R119899 with 120597ΩLipschitz the source119891 isin 119867minus1(Ω) andboundary condition119892 isin 11986712(120597Ω) It is well known that thereexists a unique solution119908 isin 1198671(Ω) such that 119908minus 119892 isin 1198671

0(Ω)

for the PDE (70) [40]Assuming that during the production process the work-

piece is contaminated by impurities and that such impuritiesare described by piecewise smooth function the inverseelectromagnetic problem consists in the identification andthe localization of the inhomogeneities as well as the functionvalues of the impurities The localization of support andthe tabulation of the inhomogeneities values can indicatepossible sources of contamination in the magnetic material

In other words the inverse problem in electromagnetismconsists in the identification of the support (shape) and thefunction values of 1205951 1205952 of the coefficient function 119906(119909)defined in (69) The voltage potential 119892 is chosen such thatit corresponds to the current measurement ℎ = (119908)

120584|120597Ω

available as a set of continuous measurement in 120597Ω Thisproblem is known in the literature as the inverse problem forthe Dirichlet-to-Neumann map [20]

With this framework the problem can be modeled by theoperator equation

1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)

where the potential profile 119892 = 119908|120597Ωisin 11986712

(Ω) is given


The authors in [4] investigated a level-set approach forsolving an inverse problem of identification of inhomo-geneities inside a nonlinear material from local measure-ments of the magnetic induction The assumption in [4] isthat part of the inhomogeneities is given by a crack localizedinside the workpiece and that outside the crack regionmagnetic conductivities are nonlinear and they depend onthe magnetic induction In other words that 120595

1= 1205831and

1205952= 1205832(|nabla119908|2

) where 1205831is the (constant) air conductivity

and 1205832= 1205832(|nabla119908|2

) is a nonlinear conductivity of theworkpiece material whose values are assumed to be knownIn [4] they also present a successful iterative algorithm andnumerical experiment However in [4] the measurementsand therefore the data are given in the wholeΩ Such amountof measurements is not reasonable in applications Moreoverthe proposed level-set algorithm is based on an optimalitycondition of a least square functional with1198671(Ω)-seminormregularization Therefore there is no guarantee of existenceof minimum for the proposed functional

Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879

119863is the Dirich-

let trace operator Moreover since 119879119863 1198671

(Ω) rarr 11986712

(120597Ω)

is linear and continuous [40] we have 119879119863(119908)11986712(120597Ω)

le

1198881199081198671(Ω)

In the following lemma we prove that the operator 1198652

satisfies the Assumption (A2)

Lemma 22 Let the operator 1198652 119863(119865) sub 119871

1

(Ω) rarr

11986712

(120597Ω) as defined in (71)Then1198652is continuous with respect

to the 1198711(Ω) topology

Proof Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080denoting the respective

solution of (63) The linearity of (70) implies that 119908119899minus 1199080isin

1198671

0(Ω) and it satisfies

nabla sdot (119906119899nabla119908119899) minus nabla sdot (119906

0nabla1199080) = 0 (72)

with homogeneous boundary conditionTherefore using theweak formulation for (72) we have

intΩ

(nabla sdot (119906119899nabla119908119899) minus nabla sdot (119906

0nabla1199080)) 120593119889119909 = 0 forall120593 isin 119867

1

0(Ω)

(73)

In particular theweak formulation holds true for120593 = 119908119899minus1199080

From the Green formula [40] and the assumption that119898 gt 0(that guarantee ellipticity of (70)) it follows that

1198981003817100381710038171003817nabla119908119899 minus nabla1199080

1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080

1003817100381710038171003817 (nabla119908119899minusnabla1199080)1003816100381610038161003816 119889119909

(74)

From [41 Theorem 1] there exists 120576 gt 0 (small enough) suchthat 119908

0isin 1198821119901

(Ω) for 119901 = 2 + 120576 Using the Holder inequality

[40] with 1119901 + 1119902 = 12 (note that 119902 gt 2 in (74)) it followsthat


1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)

1003817100381710038171003817nabla119908119899 minus nabla119908010038171003817100381710038171198712(Ω)

(75)

Therefore using the Poincare inequality [40] and (75) wehave

1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)

where the constant 119862 depends only of 119898Ω nabla1199080 and the

Poincare constant Now the assertion follows from Lemma 5and Remark 21

621 A Level-Set Algorithm for Inverse Problem in NonlinearElectromagnetism Wepropose an explicit iterative algorithmderived from the optimality conditions (57) (58) (59)(60) (61) and (62) for the Tikhonov functional G

120576120572 Each

iteration of this algorithm consists in the following steps inthe first step the residual vector 119903 isin 1198712(120597Ω) corresponding tothe iterate (120601

119899 1205951

119899 1205952

119899) is evaluated This requires the solution

of one elliptic BVP of Dirichlet type In the second stepthe solutions 119907 isin 119867

1

(Ω) of the adjoint problems for theresidual components 119903 are evaluated This corresponds tosolving one elliptic BVP of Neumann type and to computingthe inner product nabla119908 sdot nabla119907 in 1198712(Ω) Next the computationof 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and

(62) follows The fourth step is the updates of the level-setfunction 120575120601

119899isin 1198671

(Ω) and the level function values 120575120595119895119899isin

119861119881(Ω) by solving (57) (58) and (59)The algorithm is summarized in Algorithm 23

Algorithm 23 (an explicit algorithm based on the proposedlevel-set iterative regularization method)

(1) Evaluate the residual 119903 = 1198652(119875120576(120601119899 1205951

119899 1205952

119899)) minus 119910

120575

=

119908|120597Ωminus 119892120575 where 119908 isin 1198671(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast

119903 = nabla119908 sdot nabla119907 isin 1198712

(Ω)where 119908 is the function computed in step (1) and 119907 isin1198671

(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as

in (60) (61) and (62)(4) Evaluate the updates 120575120601 isin 1198671(Ω) 120575120595119895 isin 119861119881(Ω) by

solving (57) (58) and (59)(5) Update the level-set functions 120601

119899+1= 120601119899+ (1120572)120575120601

and the level function values 120595119895119899+1= 120595119895

119899+ (1120572)120575120595

119895


7 Conclusions and Future Directions

In this paper we generalize the results of convergence andstability of the level-set regularization approach proposedin [18 19] where the level values of discontinuities arenot piecewise constant inside each region We analyze theparticular case where the set Ω is divided in two regionsHowever it is easy to extend the analysis for the case ofmultiple regions adapting the multiple level-set approach in[17 19]

We apply the level-set framework for two problemsthe inverse potential problem and in an inverse problemin nonlinear electromagnetism with piecewise nonconstantsolution In both cases we prove that the parameter-to-solution map satisfies the assumption (A1) The inversepotential problem application is a natural generalization ofthe problem computed in [17ndash19] We also investigate theapplicability of an inverse problem in nonlinear electro-magnetism in the identification of inhomogeneities insidea nonlinear magnetic workpiece Moreover we proposeiterative algorithm based on the optimality condition of thesmooth Tikhonov functionalG

120576120572

A natural continuation of this paper is the numericalimplementation Level-set numerical implementations forthe inverse potential problem were done before in [17ndash19]where the level values are assumed to be constant Implemen-tations of level-set methods for resistivityconductivity prob-lem in elliptic equation have been intensively implementedrecently for example [2 4 6 10 11 42 43]

Acknowledgment

A De Cezaro acknowledges the partial support from IMEF-FURG

References

[1] F Santosa ldquoA level-set approach for inverse problems involvingobstaclesrdquo ESAIM Control Optimisation and Calculus of Vari-ations vol 1 pp 17ndash33 1996

[2] M Burger and S J Osher ldquoA survey on level set methodsfor inverse problems and optimal designrdquo European Journal ofApplied Mathematics vol 16 no 2 pp 263ndash301 2005

[3] F Chantalat C-H Bruneau C Galusinski and A IolloldquoLevel-set penalization and cartesian meshes a paradigm forinverse problems and optimal designrdquo Journal of ComputationalPhysics vol 228 no 17 pp 6291ndash6315 2009

[4] I Cimrak and R van Keer ldquoLevel set method for the inverseelliptic problem in nonlinear electromagnetismrdquo Journal ofComputational Physics vol 229 no 24 pp 9269ndash9283 2010

[5] B Dong A Chien YMao J Ye F Vinuela and S Osher ldquoLevelset based brain aneurysmcapturing in 3Drdquo Inverse Problems andImaging vol 4 no 2 pp 241ndash255 2010

[6] O Dorn and D Lesselier ldquoLevel set methods for inversescatteringmdashsome recent developmentsrdquo Inverse Problems vol25 no 12 Article ID 125001 11 pages 2009

[7] E Maitre and F Santosa ldquoLevel set methods for optimizationproblems involving geometry and constraints II Optimizationover a fixed surfacerdquo Journal of Computational Physics vol 227no 22 pp 9596ndash9611 2008

[8] B F Nielsen M Lysaker and A Tveito ldquoOn the use of theresting potential and level set methods for identifying ischemicheart disease an inverse problemrdquo Journal of ComputationalPhysics vol 220 no 2 pp 772ndash790 2007

[9] Y-H R Tsai and SOsher ldquoTotal variation and level setmethodsin image sciencerdquo Acta Numerica vol 14 pp 509ndash573 2005

[10] K van den Doel and U M Ascher ldquoOn level set regular-ization for highly ill-posed distributed parameter estimationproblemsrdquo Journal of Computational Physics vol 216 no 2 pp707ndash723 2006

[11] E T Chung T F Chan and X-C Tai ldquoElectrical impedancetomography using level set representation and total variationalregularizationrdquo Journal of Computational Physics vol 205 no1 pp 357ndash372 2005

[12] J Chung and L Vese ldquoImage segmantation using a multilayerlevel-sets apprachrdquo UCLA CAM Report vol 193 no 03ndash53pp 1ndash28 2003

[13] J Lie M Lysaker and X-C Tai ldquoA variant of the level setmethod and applications to image segmentationrdquoMathematicsof Computation vol 75 no 255 pp 1155ndash1174 2006

[14] X-C Tai andC-H Yao ldquoImage segmentation by piecewise con-stant Mumford-Shah model without estimating the constantsrdquoJournal of Computational Mathematics vol 24 no 3 pp 435ndash443 2006

[15] L A Vese and S J Osher ldquoImage denoising and decompositionwith total variation minimization and oscillatory functionsrdquoJournal of Mathematical Imaging and Vision vol 20 no 1-2 pp7ndash18 2004

[16] X-C Tai andT F Chan ldquoA survey onmultiple level setmethodswith applications for identifying piecewise constant functionsrdquoInternational Journal of Numerical Analysis amp Modeling vol 1no 1 pp 25ndash47 2004

[17] A DeCezaro A Leitao and X-C Tai ldquoOn multiple level-setregularizationmethods for inverse problemsrdquo Inverse Problemsvol 25 no 3 Article ID 035004 22 pages 2009

[18] A De Cezaro and A Leitao ldquoLevel-set approaches of 1198712-type

for recovering shape and contrast in ill-posed problemsrdquo InverseProblems in Science and Engineering vol 20 no 4 pp 571ndash5872012

[19] A De Cezaro A Leitao and X-C Tai ldquoOn a level-set typemethods for recovering picewise constant solution of ill-posedproblemsrdquo in Proceedings of the 2nd International Conference onScale Space and Variational Methods in Computer Vision (SSVMrsquo09) vol 5567 of Lecture Notes in Computer Science pp 50ndash622009

[20] V Isakov Inverse Source Problems vol 34 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 1990

[21] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Mathematics and Its Applications vol 375Kluwer Academic Publishers DodrechtTheNetherlands 1996

[22] F Hettlich and W Rundell ldquoIterative methods for the recon-struction of an inverse potential problemrdquo Inverse Problems vol12 no 3 pp 251ndash266 1996

[23] T F Chan and X-C Tai ldquoIdentification of discontinuous coeffi-cients in elliptic problems using total variation regularizationrdquoSIAM Journal on Scientific Computing vol 25 no 3 pp 881ndash904 2003

[24] T F Chan and X-C Tai ldquoLevel set and total variation reg-ularization for elliptic inverse problems with discontinuouscoefficientsrdquo Journal of Computational Physics vol 193 no 1 pp40ndash66 2004


[25] F Fruhauf O Scherzer and A Leitao ldquoAnalysis of regulariza-tion methods for the solution of ill-posed problems involvingdiscontinuous operatorsrdquo SIAM Journal on Numerical Analysisvol 43 no 2 pp 767ndash786 2005

[26] A Leitao and O Scherzer ldquoOn the relation between constraintregularization level sets and shape optimizationrdquo Inverse Prob-lems vol 19 no 1 pp L1ndashL11 2003

[27] X-C Tai and H Li ldquoA piecewise constant level set method forelliptic inverse problemsrdquo Applied Numerical Mathematics vol57 no 5-7 pp 686ndash696 2007

[28] M Burger ldquoA level set method for inverse problemsrdquo InverseProblems vol 17 no 5 pp 1327ndash1355 2001

[29] K van den Doel U M Ascher and A Leitao ldquoMultiple levelsets for piecewise constant surface reconstruction in highly ill-posed problemsrdquo Journal of Scientific Computing vol 43 no 1pp 44ndash66 2010

[30] P A Berthelsen ldquoA decomposed immersed interface methodfor variable coefficient elliptic equations with non-smooth anddiscontinuous solutionsrdquo Journal of Computational Physics vol197 no 1 pp 364ndash386 2004

[31] X-C Tai and H Li ldquoA piecewise constant level set method forelliptic inverse problemsrdquo Applied Numerical Mathematics vol57 no 5ndash7 pp 686ndash696 2007

[32] K van den Doel U M Ascher and D K Pai ldquoSourcelocalization in electromyography using the inverse potentialproblemrdquo Inverse Problems vol 27 no 2 Article ID 025008 20pages 2011

[33] L C Evans and R F Gariepy Measure Theory and FineProperties of Functions Studies in AdvancedMathematics CRCPress Boca Raton Fla USA 1992

[34] R Acar and C R Vogel ldquoAnalysis of bounded variation penaltymethods for ill-posed problemsrdquo Inverse Problems vol 10 no6 pp 1217ndash1229 1994

[35] M C Delfour and J-P Zolesio ldquoShape analysis via orienteddistance functionsrdquo Journal of Functional Analysis vol 123 no1 pp 129ndash201 1994

[36] R A Adams Sobolev Spaces Academic Press New York NYUSA 1975

[37] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of non-linear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[38] K Yosida Functional Analysis Classics in MathematicsSpringer Berlin Germany 1995 Reprint of the sixth (1980)edition

[39] K van den Doel U M Ascher and D K Pai ldquoComputed myo-graphy three-dimensional reconstruction of motor functionsfrom surface EMG datardquo Inverse Problems vol 24 no 6 ArticleID 065010 17 pages 2008

[40] R Dautray and J L LionsMathematical Analysis and Numeri-cal Methods for Science and Technology vol 2 Springer BerlinGermany 1988

[41] N G Meyers ldquoAn 119871119901-estimate for the gradient of solutions ofsecond order elliptic divergence equationsrdquo Annali della ScuolaNormale Superiore di Pisa vol 17 no 3 pp 189ndash206 1963

[42] M Soleimani W R B Lionheart and O Dorn ldquoLevel setreconstruction of conductivity and permittivity from boundaryelectrical measurements using experimental datardquo Inverse Prob-lems in Science and Engineering vol 14 no 2 pp 193ndash210 2006

[43] K van den Doel and U M Ascher ldquoDynamic level set regu-larization for large distributed parameter estimation problemsrdquoInverse Problems vol 23 no 3 pp 1271ndash1288 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


119910 isin 119884 are not known and we have only access to noise data119910120575

isin 119884 whose level of noise 120575 gt 0 is assumed to be known apriori and satisfies

10038171003817100381710038171003817119910120575

minus 11991010038171003817100381710038171003817119884le 120575 (2)

We assume that the inverse problem associated with theoperator equation (1) is ill-posed Indeed it is the case inmany interesting problems [4 6 10 16 20ndash22] Thereforeaccuracy of an approximated solution calls for a regulariza-tion method [21] In this paper we propose a Tikhonov-typeregularization method coupled with a level-set approach toobtain a stable approximation of the unknown level sets andvalues of the piecewise (not necessarily constant) solution of(1)

Many approaches in particular level-set type approacheshave previously been suggested for such problems In [111 23ndash26] level-set approaches for identification of theunknown parameter 119906 with distinct but known piecewiseconstant values were investigated

In [12 17 24] level-set approaches were derived to solveinverse problems assuming that 119906 is defined by severaldistinct constant values In both cases one needs only toidentify the level sets of 119906 that is the inverse problem reducesto a shape identification problem On the other hand whenthe level values of 119906 are also unknown the inverse problembecomes harder since we have to identify both the level setsand the level values of the unknown parameter 119906 In thissituation the dimension of the parameter space increases bythe number of unknown level values Level-set approachesto ill-posed problems with unknown constant level valuesappeared before in [14 16 18 19 27] Level-set regularizationproperties of the approximated solution for inverse problemsare described in [17ndash19 25 28]

However regularization theory for inverse problemswhere the components of the parameter 119906 are variable andhave discontinuities has not been well investigated Indeedlevel-set regularization theory applied to inverse problems[17ndash19] that recover the shape and the values of variablediscontinuous coefficients is unknown to the author Someearly results in the numerical implementation of level-set typemethods were previously used to obtain solutions of ellipticproblems with discontinuous and variable coefficients in [4]

In this paper we propose a level-set type regularizationmethod to ill-posed problems whose solution is composed bypiecewise components which are not necessarily constantsIn other words we introduce a level-set type regularizationmethod to recover the shape and the values of variablediscontinuous coefficients In this framework a level-setfunction is used to parameterize the solution 119906 of (1)We obtain a regularized solution using a Tikhonov-typeregularization method since the level values of 119906 are notconstant and also unknown

In the theoretical point of view the advantage of ourapproach in relation to [2 17ndash19 25 29] is that we areable to obtain regularized solutions to inverse problems with

piecewise solutions that are more general than those coveredby the regularizationmethods proposed beforeWe still proveregularization properties for the approximated solution ofthe inverse problem model (1) where the parameter is anonconstant piecewise solution The topologies needed toguarantee the existence of a minimizer (in a generalizedsense) of the Tikhonov functional (defined in (7)) are quitecomplicated and differ in some key points from [18 19 25]In this particular approach the definition of generalizedminimizers is quite different from other works [17 19 25](see Definition 3) As a consequence the arguments usedto prove the well-posedness of the Tikhonov functionalthe stability and convergence of the regularized solutionsof the inverse problem (1) are quite complicated and needsignificant improvements (see Section 3)

The main applicability advantage of the proposed level-set type method compared to that in the literature is that weare able to apply this method to problems whose solutionsdepend of nonconstant parameters This implies that weare able to handle more general and interesting physicalproblems where the components of the desired parameterare not necessarily homogeneous as those presented beforein the literature [4 6 14 16 18 19 27 30ndash32] Examplesof such interesting physical problems are heat conductionbetween materials of different heat capacity and conductiv-ity interface diffusion processes and many other types ofphysical problems where modeling components are relatedwith embedded boundaries See for example [3 4 6 1930 32] and references therein As a benchmark problem weanalyze two inverse problems modeled by elliptic PDEs withdiscontinuous and variable coefficients

In contrast the nonconstant characteristics of the levelvalues impose different types of theoretical problems sincethe topologies where we are able to provide regularizationproperties of the approximated solution are more compli-cated than the ones presented before [14 16 18 19 27]As a consequence the numerical implementations becomeharder than the other approaches in the literature [18 19 2932]

This paper is outlined as follows in Section 2 weformulate the Tikhonov functional based on the level-setframework In Section 3 we present the general assump-tions needed in this paper and the definition of the setof admissible solutions We prove relevant properties aboutthe admissible set of solutions in particular convergencein suitable topologies We also present relevant propertiesof the penalization functional In Section 4 we prove thatthe proposed method is a regularization method to inverseproblems that is we prove that the minimizers of theproposed Tikhonov functional are stable and convergentwith respect to the noise level in the data In Section 5 asmooth functional is proposed to approximate minimizersof the Tikhonov functional defined in the admissible setof solutions We provide approximation properties and theoptimality condition for the minimizers of the smoothTikhonov functional In Section 6 we present an applicationof the proposed framework to solve some interesting inverseelliptic problems with variable coefficients Conclusions andfuture directions are presented in Section 7




(119909) 1205952


2






119867 1198671

(Ω) 997888rarr 119871infin

(Ω)


(3)


119906 (119909) = 1205951

(119909)119867 (120601) + 1205952

(119909) (1 minus 119867 (120601))

= 119875 (120601 1205951

1205952

) (119909)

(4)





intΩ


1003816100381610038161003816nabla sdot (120593) (119909)1003816100381610038161003816 119889119909

le |119872|1003817100381710038171003817nabla sdot 120593

10038171003817100381710038171198711(Ω) forall120593 isin 119862

1

0(ΩR

119899

)

(5)

Hence 119891 isin 119861119881(Ω)



119865 (119875 (120601 1205951

1205952

)) = 119910120575

(6)


1205952



G120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875 (120601 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572

1205731

1003816100381610038161003816119867 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881

(7)

where 1206010and 120595119895



119894 119894 = 1 2 3





119894 119894 = 1 2 3 is provided in








(Ω) into the set



119863 (119865) = 119906 = 119902 (119907 1205951

1205952

) 119907 isinV 1205951

1205952

isin B (9)

where

119902 V times B2

ni (119907 1205951

1205952

) 997891997888rarr 1205951

119907 + 1205952

(1 minus 119907) isin 119861119881 (Ω)

(10)







lowastisin 1198671

(Ω)


lowast= 0

such that 119867(120601lowast) = 119911





119902(119911lowast 1205951

lowast 1205952



119875120576(120601 1205951

1205952

) = 1205951

119867120576(120601) + 120595

2

(1 minus 119867120576(120601)) (11)


119867120576(119905) =

1 +119905

120576for 119905 isin [minus120576 0]

119867 (119905) for 119905 isin R

[minus120576 0]

(12)


(Ω)119867120576(120601119896)






120572has a closed



120576be defined






120601119896minus 1206011198712(Ω)= 0



119867120576119896

(120601119896) minus 119911

1198711(Ω)= 0




119896

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (13)



G120572(119911 120601 120595

1

1205952

) =10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

)

(14)


119877 (119911 120601 1205951

1205952

) = 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881 (15)



[1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)]

(16)


119896


The convergence |120595119895119896|119861119881

rarr |120595119895

|119861119881






119896= 120595119895 and




(120601119896) minus 119911

100381710038171003817100381710038171198711(Ω)=10038171003817100381710038171003817119867120576119896

(120601119896) minus 119867 (120601)

100381710038171003817100381710038171198711(Ω)

= int(120601)minus1[minus1205761198960]

10038161003816100381610038161003816100381610038161003816

1 minus120601

120576119896

10038161003816100381610038161003816100381610038161003816

119889119909

le int

0

minus120576119896

int(120601)minus1(120591)

1119889120591

le meas (120601)minus1 (120591) int0

minus120576119896

1119889119905 997888rarr 0

(17)


119896119896isinN in




120576








119867120576(120601119896)





119896isin [119898119872]




119896 1205952


119902(119911 1205951

1205952

) in 1198711(Ω)



1205952

) rarr 119902(119911 1205951

1205952





119896119896isinN


120576(120601119896) 1205951

119896 1205952

119896) rarr

119902(119867120576(120601) 120595

1

1205952

) in 1198711(Ω) as 119896 rarr infin



119896 1205952

119896) minus 119902 (119911 120595

1

1205952

)100381710038171003817100381710038171198711(Ω)

=100381710038171003817100381710038171205951

119896119911119896+ 1205952

119896(1 minus 119911

119896) minus 1205951

119911 minus 1205952

(1 minus 119911)100381710038171003817100381710038171198711(Ω)

le10038171003817100381710038171199111198961003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205951

119896minus 1205951100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205951100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

+10038171003817100381710038171 minus 119911119896

1003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205952

119896minus 1205952100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205952100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

119896rarrinfin

997888rarr 0

(18)



120576(120601) isin 119871

infin

(Ω) for all 120576 gt 0As 119867

120576(120601119896) minus 119867


119896minus 1206011198712(Ω)







120595119895

119896119897

119897rarrinfin

997888rarr 120595119895

119896in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(19)



120595119895

119896119897(119896)

119896rarrinfin

997888rarr 120595119895 in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897(119896)

10038161003816100381610038161003816119861119881

119896rarrinfin

997888rarr1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881

(20)





isin 119861119881


(Ω))2 topology

Lemma 8 Let (119911119896 120601119896 1205951

119896 1205952



1

1205952



119896 1205951

119896 1205952

119896)


119896119897119897isinN in 119867

1

(Ω) 1205951119896119897119897isinN 120595

2

119896119897119897isinN

in 119861119881 times 119862infin



120601119896119897

119897rarrinfin

997888rarr 120601119896

in 1198712 (Ω)

119867120576119897

119896

(120601119896119897)119897rarrinfin

997888rarr 119911119896

in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(21)



120576120591(119896)

119896le1

2120576120591(119896minus1)

119896minus1

1003817100381710038171003817120601119896120591(119896) minus 12060111989610038171003817100381710038171198712(Ω)

le1

119896

100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)le1

119896

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(22)


10038171003817100381710038171198712(Ω)

le1003817100381710038171003817120601 minus 120601119896

10038171003817100381710038171198712(Ω)+1003817100381710038171003817120601119896120591(119896) minus 120601119896

10038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

le1003817100381710038171003817119911 minus 119911119896

10038171003817100381710038171198711(Ω)+100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)

(23)

From (22)

lim119896rarrinfin

1003817100381710038171003817120601 minus 120601119896120591(119896)10038171003817100381710038171198712(Ω)

= 0

lim119896rarrinfin

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

= 0

(24)


10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (25)




120576120591(119896)

119896

(120601119896120591(119896)


100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

ge100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω1015840)

ge 12057410038161003816100381610038161003816Ω101584010038161003816100381610038161003816 119896 isin N

(26)







119896119896isinN 120601119896119896isinN and 120595

119895



119877 (119911 120601 1205951

1205952


1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881


119896

120601119895

119896119897 isin 119867

1


119877 (119911 120601 1205951

1205952

)



1205731

100381610038161003816100381610038161003816119867120576119897

119896

(120601119896119897)100381610038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896119897 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896119897minus 120595119895

0

10038161003816100381610038161003816119861119881





119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)


120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)


(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)



119896 120601119896 1205951

119896 1205952


119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)


119877 (119911 120601 1205951

1205952


119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)




infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)


119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr


119888(ΩR119899) |120585| le 1 Then


intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]


[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)


Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)


(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)








119896 120601119896 1205951

119896 1205952

119896) be a



119896 1205952

119896) rarr inf G

120572le


120572(119911119896 120601119896 1205951

119896 1205952



119896 1205952

119896) is uni-


119896 120601119896 1205951

119896 1205952

119896)) and the


119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1




G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)



(119911120572 120601120572 1205951

120572 1205952







1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119909) 1205952


2






119867 1198671

(Ω) 997888rarr 119871infin

(Ω)


(3)


119906 (119909) = 1205951

(119909)119867 (120601) + 1205952

(119909) (1 minus 119867 (120601))

= 119875 (120601 1205951

1205952

) (119909)

(4)





intΩ


1003816100381610038161003816nabla sdot (120593) (119909)1003816100381610038161003816 119889119909

le |119872|1003817100381710038171003817nabla sdot 120593

10038171003817100381710038171198711(Ω) forall120593 isin 119862

1

0(ΩR

119899

)

(5)

Hence 119891 isin 119861119881(Ω)



119865 (119875 (120601 1205951

1205952

)) = 119910120575

(6)


1205952



G120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875 (120601 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572

1205731

1003816100381610038161003816119867 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881

(7)

where 1206010and 120595119895



119894 119894 = 1 2 3





119894 119894 = 1 2 3 is provided in








(Ω) into the set



119863 (119865) = 119906 = 119902 (119907 1205951

1205952

) 119907 isinV 1205951

1205952

isin B (9)

where

119902 V times B2

ni (119907 1205951

1205952

) 997891997888rarr 1205951

119907 + 1205952

(1 minus 119907) isin 119861119881 (Ω)

(10)







lowastisin 1198671

(Ω)


lowast= 0

such that 119867(120601lowast) = 119911





119902(119911lowast 1205951

lowast 1205952



119875120576(120601 1205951

1205952

) = 1205951

119867120576(120601) + 120595

2

(1 minus 119867120576(120601)) (11)


119867120576(119905) =

1 +119905

120576for 119905 isin [minus120576 0]

119867 (119905) for 119905 isin R

[minus120576 0]

(12)


(Ω)119867120576(120601119896)






120572has a closed



120576be defined






120601119896minus 1206011198712(Ω)= 0



119867120576119896

(120601119896) minus 119911

1198711(Ω)= 0




119896

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (13)



G120572(119911 120601 120595

1

1205952

) =10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

)

(14)


119877 (119911 120601 1205951

1205952

) = 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881 (15)



[1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)]

(16)


119896


The convergence |120595119895119896|119861119881

rarr |120595119895

|119861119881






119896= 120595119895 and




(120601119896) minus 119911

100381710038171003817100381710038171198711(Ω)=10038171003817100381710038171003817119867120576119896

(120601119896) minus 119867 (120601)

100381710038171003817100381710038171198711(Ω)

= int(120601)minus1[minus1205761198960]

10038161003816100381610038161003816100381610038161003816

1 minus120601

120576119896

10038161003816100381610038161003816100381610038161003816

119889119909

le int

0

minus120576119896

int(120601)minus1(120591)

1119889120591

le meas (120601)minus1 (120591) int0

minus120576119896

1119889119905 997888rarr 0

(17)


119896119896isinN in




120576








119867120576(120601119896)





119896isin [119898119872]




119896 1205952


119902(119911 1205951

1205952

) in 1198711(Ω)



1205952

) rarr 119902(119911 1205951

1205952





119896119896isinN


120576(120601119896) 1205951

119896 1205952

119896) rarr

119902(119867120576(120601) 120595

1

1205952

) in 1198711(Ω) as 119896 rarr infin



119896 1205952

119896) minus 119902 (119911 120595

1

1205952

)100381710038171003817100381710038171198711(Ω)

=100381710038171003817100381710038171205951

119896119911119896+ 1205952

119896(1 minus 119911

119896) minus 1205951

119911 minus 1205952

(1 minus 119911)100381710038171003817100381710038171198711(Ω)

le10038171003817100381710038171199111198961003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205951

119896minus 1205951100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205951100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

+10038171003817100381710038171 minus 119911119896

1003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205952

119896minus 1205952100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205952100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

119896rarrinfin

997888rarr 0

(18)



120576(120601) isin 119871

infin

(Ω) for all 120576 gt 0As 119867

120576(120601119896) minus 119867


119896minus 1206011198712(Ω)







120595119895

119896119897

119897rarrinfin

997888rarr 120595119895

119896in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(19)



120595119895

119896119897(119896)

119896rarrinfin

997888rarr 120595119895 in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897(119896)

10038161003816100381610038161003816119861119881

119896rarrinfin

997888rarr1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881

(20)





isin 119861119881


(Ω))2 topology

Lemma 8 Let (119911119896 120601119896 1205951

119896 1205952



1

1205952



119896 1205951

119896 1205952

119896)


119896119897119897isinN in 119867

1

(Ω) 1205951119896119897119897isinN 120595

2

119896119897119897isinN

in 119861119881 times 119862infin



120601119896119897

119897rarrinfin

997888rarr 120601119896

in 1198712 (Ω)

119867120576119897

119896

(120601119896119897)119897rarrinfin

997888rarr 119911119896

in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(21)



120576120591(119896)

119896le1

2120576120591(119896minus1)

119896minus1

1003817100381710038171003817120601119896120591(119896) minus 12060111989610038171003817100381710038171198712(Ω)

le1

119896

100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)le1

119896

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(22)


10038171003817100381710038171198712(Ω)

le1003817100381710038171003817120601 minus 120601119896

10038171003817100381710038171198712(Ω)+1003817100381710038171003817120601119896120591(119896) minus 120601119896

10038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

le1003817100381710038171003817119911 minus 119911119896

10038171003817100381710038171198711(Ω)+100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)

(23)

From (22)

lim119896rarrinfin

1003817100381710038171003817120601 minus 120601119896120591(119896)10038171003817100381710038171198712(Ω)

= 0

lim119896rarrinfin

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

= 0

(24)


10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (25)




120576120591(119896)

119896

(120601119896120591(119896)


100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

ge100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω1015840)

ge 12057410038161003816100381610038161003816Ω101584010038161003816100381610038161003816 119896 isin N

(26)







119896119896isinN 120601119896119896isinN and 120595

119895



119877 (119911 120601 1205951

1205952


1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881


119896

120601119895

119896119897 isin 119867

1


119877 (119911 120601 1205951

1205952

)



1205731

100381610038161003816100381610038161003816119867120576119897

119896

(120601119896119897)100381610038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896119897 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896119897minus 120595119895

0

10038161003816100381610038161003816119861119881





119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)


120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)


(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)



119896 120601119896 1205951

119896 1205952


119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)


119877 (119911 120601 1205951

1205952


119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)




infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)


119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr


119888(ΩR119899) |120585| le 1 Then


intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]


[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)


Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)


(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)








119896 120601119896 1205951

119896 1205952

119896) be a



119896 1205952

119896) rarr inf G

120572le


120572(119911119896 120601119896 1205951

119896 1205952



119896 1205952

119896) is uni-


119896 120601119896 1205951

119896 1205952

119896)) and the


119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1




G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)



(119911120572 120601120572 1205951

120572 1205952







1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(Ω) into the set



119863 (119865) = 119906 = 119902 (119907 1205951

1205952

) 119907 isinV 1205951

1205952

isin B (9)

where

119902 V times B2

ni (119907 1205951

1205952

) 997891997888rarr 1205951

119907 + 1205952

(1 minus 119907) isin 119861119881 (Ω)

(10)







lowastisin 1198671

(Ω)


lowast= 0

such that 119867(120601lowast) = 119911





119902(119911lowast 1205951

lowast 1205952



119875120576(120601 1205951

1205952

) = 1205951

119867120576(120601) + 120595

2

(1 minus 119867120576(120601)) (11)


119867120576(119905) =

1 +119905

120576for 119905 isin [minus120576 0]

119867 (119905) for 119905 isin R

[minus120576 0]

(12)


(Ω)119867120576(120601119896)






120572has a closed



120576be defined






120601119896minus 1206011198712(Ω)= 0



119867120576119896

(120601119896) minus 119911

1198711(Ω)= 0




119896

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (13)



G120572(119911 120601 120595

1

1205952

) =10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

)

(14)


119877 (119911 120601 1205951

1205952

) = 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881 (15)



[1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)]

(16)


119896


The convergence |120595119895119896|119861119881

rarr |120595119895

|119861119881






119896= 120595119895 and




(120601119896) minus 119911

100381710038171003817100381710038171198711(Ω)=10038171003817100381710038171003817119867120576119896

(120601119896) minus 119867 (120601)

100381710038171003817100381710038171198711(Ω)

= int(120601)minus1[minus1205761198960]

10038161003816100381610038161003816100381610038161003816

1 minus120601

120576119896

10038161003816100381610038161003816100381610038161003816

119889119909

le int

0

minus120576119896

int(120601)minus1(120591)

1119889120591

le meas (120601)minus1 (120591) int0

minus120576119896

1119889119905 997888rarr 0

(17)


119896119896isinN in




120576








119867120576(120601119896)





119896isin [119898119872]




119896 1205952


119902(119911 1205951

1205952

) in 1198711(Ω)



1205952

) rarr 119902(119911 1205951

1205952





119896119896isinN


120576(120601119896) 1205951

119896 1205952

119896) rarr

119902(119867120576(120601) 120595

1

1205952

) in 1198711(Ω) as 119896 rarr infin



119896 1205952

119896) minus 119902 (119911 120595

1

1205952

)100381710038171003817100381710038171198711(Ω)

=100381710038171003817100381710038171205951

119896119911119896+ 1205952

119896(1 minus 119911

119896) minus 1205951

119911 minus 1205952

(1 minus 119911)100381710038171003817100381710038171198711(Ω)

le10038171003817100381710038171199111198961003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205951

119896minus 1205951100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205951100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

+10038171003817100381710038171 minus 119911119896

1003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205952

119896minus 1205952100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205952100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

119896rarrinfin

997888rarr 0

(18)



120576(120601) isin 119871

infin

(Ω) for all 120576 gt 0As 119867

120576(120601119896) minus 119867


119896minus 1206011198712(Ω)







120595119895

119896119897

119897rarrinfin

997888rarr 120595119895

119896in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(19)



120595119895

119896119897(119896)

119896rarrinfin

997888rarr 120595119895 in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897(119896)

10038161003816100381610038161003816119861119881

119896rarrinfin

997888rarr1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881

(20)





isin 119861119881


(Ω))2 topology

Lemma 8 Let (119911119896 120601119896 1205951

119896 1205952



1

1205952



119896 1205951

119896 1205952

119896)


119896119897119897isinN in 119867

1

(Ω) 1205951119896119897119897isinN 120595

2

119896119897119897isinN

in 119861119881 times 119862infin



120601119896119897

119897rarrinfin

997888rarr 120601119896

in 1198712 (Ω)

119867120576119897

119896

(120601119896119897)119897rarrinfin

997888rarr 119911119896

in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(21)



120576120591(119896)

119896le1

2120576120591(119896minus1)

119896minus1

1003817100381710038171003817120601119896120591(119896) minus 12060111989610038171003817100381710038171198712(Ω)

le1

119896

100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)le1

119896

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(22)


10038171003817100381710038171198712(Ω)

le1003817100381710038171003817120601 minus 120601119896

10038171003817100381710038171198712(Ω)+1003817100381710038171003817120601119896120591(119896) minus 120601119896

10038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

le1003817100381710038171003817119911 minus 119911119896

10038171003817100381710038171198711(Ω)+100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)

(23)

From (22)

lim119896rarrinfin

1003817100381710038171003817120601 minus 120601119896120591(119896)10038171003817100381710038171198712(Ω)

= 0

lim119896rarrinfin

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

= 0

(24)


10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (25)




120576120591(119896)

119896

(120601119896120591(119896)


100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

ge100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω1015840)

ge 12057410038161003816100381610038161003816Ω101584010038161003816100381610038161003816 119896 isin N

(26)







119896119896isinN 120601119896119896isinN and 120595

119895



119877 (119911 120601 1205951

1205952


1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881


119896

120601119895

119896119897 isin 119867

1


119877 (119911 120601 1205951

1205952

)



1205731

100381610038161003816100381610038161003816119867120576119897

119896

(120601119896119897)100381610038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896119897 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896119897minus 120595119895

0

10038161003816100381610038161003816119861119881





119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)


120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)


(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)



119896 120601119896 1205951

119896 1205952


119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)


119877 (119911 120601 1205951

1205952


119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)




infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)


119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr


119888(ΩR119899) |120585| le 1 Then


intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]


[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)


Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)


(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)








119896 120601119896 1205951

119896 1205952

119896) be a



119896 1205952

119896) rarr inf G

120572le


120572(119911119896 120601119896 1205951

119896 1205952



119896 1205952

119896) is uni-


119896 120601119896 1205951

119896 1205952

119896)) and the


119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1




G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)



(119911120572 120601120572 1205951

120572 1205952







1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119896= 120595119895 and




(120601119896) minus 119911

100381710038171003817100381710038171198711(Ω)=10038171003817100381710038171003817119867120576119896

(120601119896) minus 119867 (120601)

100381710038171003817100381710038171198711(Ω)

= int(120601)minus1[minus1205761198960]

10038161003816100381610038161003816100381610038161003816

1 minus120601

120576119896

10038161003816100381610038161003816100381610038161003816

119889119909

le int

0

minus120576119896

int(120601)minus1(120591)

1119889120591

le meas (120601)minus1 (120591) int0

minus120576119896

1119889119905 997888rarr 0

(17)


119896119896isinN in




120576








119867120576(120601119896)





119896isin [119898119872]




119896 1205952


119902(119911 1205951

1205952

) in 1198711(Ω)



1205952

) rarr 119902(119911 1205951

1205952





119896119896isinN


120576(120601119896) 1205951

119896 1205952

119896) rarr

119902(119867120576(120601) 120595

1

1205952

) in 1198711(Ω) as 119896 rarr infin



119896 1205952

119896) minus 119902 (119911 120595

1

1205952

)100381710038171003817100381710038171198711(Ω)

=100381710038171003817100381710038171205951

119896119911119896+ 1205952

119896(1 minus 119911

119896) minus 1205951

119911 minus 1205952

(1 minus 119911)100381710038171003817100381710038171198711(Ω)

le10038171003817100381710038171199111198961003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205951

119896minus 1205951100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205951100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

+10038171003817100381710038171 minus 119911119896

1003817100381710038171003817119871infin(Ω)

100381710038171003817100381710038171205952

119896minus 1205952100381710038171003817100381710038171198711(Ω)

+100381710038171003817100381710038171205952100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817119911119896 minus 11991110038171003817100381710038171198712(Ω)

119896rarrinfin

997888rarr 0

(18)



120576(120601) isin 119871

infin

(Ω) for all 120576 gt 0As 119867

120576(120601119896) minus 119867


119896minus 1206011198712(Ω)







120595119895

119896119897

119897rarrinfin

997888rarr 120595119895

119896in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(19)



120595119895

119896119897(119896)

119896rarrinfin

997888rarr 120595119895 in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897(119896)

10038161003816100381610038161003816119861119881

119896rarrinfin

997888rarr1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881

(20)





isin 119861119881


(Ω))2 topology

Lemma 8 Let (119911119896 120601119896 1205951

119896 1205952



1

1205952



119896 1205951

119896 1205952

119896)


119896119897119897isinN in 119867

1

(Ω) 1205951119896119897119897isinN 120595

2

119896119897119897isinN

in 119861119881 times 119862infin



120601119896119897

119897rarrinfin

997888rarr 120601119896

in 1198712 (Ω)

119867120576119897

119896

(120601119896119897)119897rarrinfin

997888rarr 119911119896

in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(21)



120576120591(119896)

119896le1

2120576120591(119896minus1)

119896minus1

1003817100381710038171003817120601119896120591(119896) minus 12060111989610038171003817100381710038171198712(Ω)

le1

119896

100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)le1

119896

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(22)


10038171003817100381710038171198712(Ω)

le1003817100381710038171003817120601 minus 120601119896

10038171003817100381710038171198712(Ω)+1003817100381710038171003817120601119896120591(119896) minus 120601119896

10038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

le1003817100381710038171003817119911 minus 119911119896

10038171003817100381710038171198711(Ω)+100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)

(23)

From (22)

lim119896rarrinfin

1003817100381710038171003817120601 minus 120601119896120591(119896)10038171003817100381710038171198712(Ω)

= 0

lim119896rarrinfin

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

= 0

(24)


10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (25)




120576120591(119896)

119896

(120601119896120591(119896)


100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

ge100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω1015840)

ge 12057410038161003816100381610038161003816Ω101584010038161003816100381610038161003816 119896 isin N

(26)







119896119896isinN 120601119896119896isinN and 120595

119895



119877 (119911 120601 1205951

1205952


1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881


119896

120601119895

119896119897 isin 119867

1


119877 (119911 120601 1205951

1205952

)



1205731

100381610038161003816100381610038161003816119867120576119897

119896

(120601119896119897)100381610038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896119897 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896119897minus 120595119895

0

10038161003816100381610038161003816119861119881





119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)


120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)


(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)



119896 120601119896 1205951

119896 1205952


119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)


119877 (119911 120601 1205951

1205952


119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)




infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)


119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr


119888(ΩR119899) |120585| le 1 Then


intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]


[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)


Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)


(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)








119896 120601119896 1205951

119896 1205952

119896) be a



119896 1205952

119896) rarr inf G

120572le


120572(119911119896 120601119896 1205951

119896 1205952



119896 1205952

119896) is uni-


119896 120601119896 1205951

119896 1205952

119896)) and the


119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1




G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)



(119911120572 120601120572 1205951

120572 1205952







1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











isin 119861119881


(Ω))2 topology

Lemma 8 Let (119911119896 120601119896 1205951

119896 1205952



1

1205952



119896 1205951

119896 1205952

119896)


119896119897119897isinN in 119867

1

(Ω) 1205951119896119897119897isinN 120595

2

119896119897119897isinN

in 119861119881 times 119862infin



120601119896119897

119897rarrinfin

997888rarr 120601119896

in 1198712 (Ω)

119867120576119897

119896

(120601119896119897)119897rarrinfin

997888rarr 119911119896

in 1198711 (Ω)

10038161003816100381610038161003816120595119895

119896119897

10038161003816100381610038161003816119861119881

119897rarrinfin

997888rarr10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(21)



120576120591(119896)

119896le1

2120576120591(119896minus1)

119896minus1

1003817100381710038171003817120601119896120591(119896) minus 12060111989610038171003817100381710038171198712(Ω)

le1

119896

100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)le1

119896

10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881 119895 = 1 2

(22)


10038171003817100381710038171198712(Ω)

le1003817100381710038171003817120601 minus 120601119896

10038171003817100381710038171198712(Ω)+1003817100381710038171003817120601119896120591(119896) minus 120601119896

10038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

le1003817100381710038171003817119911 minus 119911119896

10038171003817100381710038171198711(Ω)+100381710038171003817100381710038171003817119867120576120591(119896)

119896

(120601119896120591(119896)

) minus 119911119896

1003817100381710038171003817100381710038171198711(Ω)

(23)

From (22)

lim119896rarrinfin

1003817100381710038171003817120601 minus 120601119896120591(119896)10038171003817100381710038171198712(Ω)

= 0

lim119896rarrinfin

100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

= 0

(24)


10038161003816100381610038161003816120595119895

119896120591(119896)

10038161003816100381610038161003816119861119881997888rarr

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881 119895 = 1 2 (25)




120576120591(119896)

119896

(120601119896120591(119896)


100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω)

ge100381710038171003817100381710038171003817119911 minus 119867

120576120591(119896)

119896

(120601119896120591(119896)

)1003817100381710038171003817100381710038171198711(Ω1015840)

ge 12057410038161003816100381610038161003816Ω101584010038161003816100381610038161003816 119896 isin N

(26)







119896119896isinN 120601119896119896isinN and 120595

119895



119877 (119911 120601 1205951

1205952


1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881


119896

120601119895

119896119897 isin 119867

1


119877 (119911 120601 1205951

1205952

)



1205731

100381610038161003816100381610038161003816119867120576119897

119896

(120601119896119897)100381610038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896119897 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896119897minus 120595119895

0

10038161003816100381610038161003816119861119881





119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)


120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)


(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)



119896 120601119896 1205951

119896 1205952


119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)


119877 (119911 120601 1205951

1205952


119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)




infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)


119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr


119888(ΩR119899) |120585| le 1 Then


intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]


[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)


Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)


(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)








119896 120601119896 1205951

119896 1205952

119896) be a



119896 1205952

119896) rarr inf G

120572le


120572(119911119896 120601119896 1205951

119896 1205952



119896 1205952

119896) is uni-


119896 120601119896 1205951

119896 1205952

119896)) and the


119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1




G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)



(119911120572 120601120572 1205951

120572 1205952







1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119877 (119911 120601 1205951

1205952

)

ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205732

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(29)


120588 (119911 120601) ge (1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)) (30)


(1205731|119911|119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671(Ω)+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

le 120588 (119911 120601) + 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881= 119877 (119911 120601 120595

1

1205952

)

(31)



119896 120601119896 1205951

119896 1205952


119911119896rarr 119911 in 1198711(Ω) 120601

119896 120601 in1198671(Ω) and 120595119895

119896rarr 120595119895 in 1198711(Ω)


119877 (119911 120601 1205951

1205952


119877 (119911119896 120601119896 1205951

119896 1205952

119896) (32)




infin

(Ω) suchthat 120595119895

119896119897minus 120595119895

1198961198711(Ω)


119896119897(119896) of 120595119895

119896119897 such that 120595119895

119896119897(119896) rarr


119888(ΩR119899) |120585| le 1 Then


intΩ

120595119895

nabla sdot 120585119889119909


intΩ

120595119895

119896119897(119896)nabla sdot 120585119889119909


[intΩ

(120595119895

119896119897(119896)minus 120595119895

119896) nabla sdot 120585119889119909 + int

Ω

120595119895

119896nabla sdot 120585119889119909]


[10038171003817100381710038171003817120595119895

119896119897(119896)minus 120595119895

119896

100381710038171003817100381710038171198711(Ω)


Ω|

minus intΩ

120585 sdot 12059011989611988910038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881]


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(33)


(see [33]) we have

1003816100381610038161003816100381612059511989510038161003816100381610038161003816119861119881= sup int

Ω

120595119895

nabla sdot 120585119889119909 120585 isin 1198621

119888(ΩR

119899

) 10038161003816100381610038161205851003816100381610038161003816 le 1


10038161003816100381610038161003816120595119895

119896

10038161003816100381610038161003816119861119881

(34)








119896 120601119896 1205951

119896 1205952

119896) be a



119896 1205952

119896) rarr inf G

120572le


120572(119911119896 120601119896 1205951

119896 1205952



119896 1205952

119896) is uni-


119896 120601119896 1205951

119896 1205952

119896)) and the


119896 120601 in1198671(Ω) 119911

119896rarr 119911 in 1198711(Ω) and

120595119895

119896rarr 120595119895 in 119871

1




G120572(119911119896 120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911119896 120601119896 1205951

119896 1205952

119896)

ge10038171003817100381710038171003817119865 (119902 (119911 120595

1

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911 120601 1205951

1205952

) = G120572(119911 120601 120595

1

1205952

)

(35)



(119911120572 120601120572 1205951

120572 1205952







1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1205952


1205952

)) = 119910



119896 120601119896 1205951

119896 1205952

119896)119896isinN of


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

119877 (119911119896 120601119896 1205951

119896 1205952

119896) 997888rarr 119898119904 lt infin

(36)


119895

119896119895=12


1198671


120601119896997888rarr 120601


119911119896997888rarr 119911dagger in 1198711 (Ω)

120595119895

119896997888rarr 120595

119895dagger in 1198711 (Ω) 119895 = 1 2

(37)



119877 (119911119896 120601119896 1205951

119896 1205952

119896) ge 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(38)


119902 (119911dagger

1205951dagger

1205951dagger


119902 (119911119896 1205951

119896 1205952

119896)

119865 (119902 (119911dagger

1205951dagger

1205952dagger


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910

(39)




120572 120601120572

1205951

120572 1205952





120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896119896isinN be


119896 120601119896 1205951

119896 1205952

119896) = (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896


120572119896


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205952dagger

)) minus 11991010038171003817100381710038171003817

+ 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

= 120572119896119877 (119911dagger

120601dagger

1205951dagger

1205952dagger

)

(40)

Since120572119896119877(119911119896 120601119896 1205951

119896 1205952

119896) le 119866120572119896

(119911119896 120601119896 1205951

119896 1205952


(40) that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) lt infin (41)


follows that

lim119896rarrinfin

120572119896119877 (119911dagger

120601dagger

1205951dagger

1205951dagger

) = 0 (42)


119911119896 and 120595119895



(119911119896 120601119896 1205951

119896 1205952

119896)

997888rarr ( 120601 1

2


(43)


lim119896rarrinfin

119902(119911119896 1205951

119896 1205952




119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119865 (119902 (

1

2

)) (44)


119877 ( 120601 1

2


119877 (119911119896 120601119896 1205951

119896 1205952

119896)


119877 (119911119896 120601119896 1205951

119896 1205952

119896)

le (119911dagger

120601dagger

1

2

)

(45)




120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572(120575) = 0 and lim120575rarr0

1205752

120572(120575)minus1



119896 and a

sequence 120572119896= 120572(120575

119896) such that (119911

120572119896

120601120572119896

1205951

120572119896

1205952

120572119896



119896 120601119896 1205951

119896 1205952

119896) = (119911

120572(120575119896) 120601120572(120575119896) 1205951

120572(120575119896) 1205952

120572(120575119896)) be


119866120572119896

(119911119896 120601119896 1205951

119896 1205952

119896) le

10038171003817100381710038171003817119865 (119902 (119911

dagger

1205951dagger

1205951dagger

)) minus 11991012057511989610038171003817100381710038171003817

2

119884

+ 120572 (120575119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

le 1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(46)


it follows that

119877 (119911119896 120601119896 1205951

119896 1205952

119896) le

1205752

119896

120572 (120575119896)+ 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

) (47)


119896) that

lim119896rarrinfin

10038171003817100381710038171003817119865 (119902 (119911

119896 1205951

119896 1205952

119896)) minus 119910

12057511989610038171003817100381710038171003817


(1205752

119896+ 120572 (120575

119896) 119877 (119911

dagger

120601dagger

1205951dagger

1205952dagger

)) = 0


119877 (119911119896 120601119896 1205951

119896 1205952

119896) le (119911

dagger

120601dagger

1205951dagger

1205952dagger

)

(48)


119896 1205952



119865 (119902 (119911 120601 1205951

1205952


119865 (119902 (119911119896 1205951

119896 1205952

119896)) = 119910 (49)



120572 In the






G120576120572(120601 1205951

1205952

) =10038171003817100381710038171003817119865 (119875120576(120601 1205951

1205952

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572(1205731

1003816100381610038161003816119867120576 (120601)1003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2

1198671

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(50)

where 119875120576(120601 1205951

1205952

) = 119902(119867120576(120601) 120595

1

1205952





(Ω) and 1205951198950isin B 119895 =



(Ω) times (119861119881)2

Proof Since infG120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times

(119861119881)2


sequence (120601119896 1205951

119896 1205952


lim119896rarrinfin

G120576120572(120601119896 1205951

119896 1205952

119896)

= inf G120576120572(120601 1205951

1205952

) (120601 1205951

1205952

) isin 1198671

(Ω) times B2

(51)



119896 and 120595119895

119896119895=12 are bounded in 1198671(Ω)


119896 120601 in 1198671(Ω) and from [33

Theorem 4 p 176] 120595119895119896rarr 120595



it follows that

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

2


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671

10038161003816100381610038161003816120595119895

minus 120595119895

0


10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881

(52)


119896 (that we






G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











G120576120572(120601 1205951

1205952


10038171003817100381710038171003817119865(119875120576(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


1003816100381610038161003816119867120576 (120601119896)1003816100381610038161003816119861119881


1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)


2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)


G120576120572(120601119896 1205951

119896 1205952

119896) = inf G

120576120572

(53)






120572



120576120572 1205952

120576120572)




(119867120576119896

(120601120576119896120572) 120601120576119896120572 1205951

120576119896120572 1205952


1198712

(Ω) times (1198711


120572on the set of


Proof Let (119911120572 120601120572 1205951

120572 1205952





119896rarr 120601120572in 1198712(Ω) 119867

120576119896

(120601119896) rarr 119911

120572


119888(Ω) such

that |120595119895119896|119861119881rarr |120595

119895


Lemma 9) that

119877 (119911120572 120601120572 1205951

120572 1205952

120572)


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

(54)

Let (120601120576119896

1205951

120576119896

1205952

120576119896


120576119896

1205951

120576119896

1205952

120576119896


(120601120576119896

) 120601120576119896

and 120595119895120576119896





rarr 120601 in 1198712(Ω)119867120576119896

(120601120576119896

) rarr

in1198711(Ω) and120595119895120576119896

rarr 119895 in1198711(Ω) 119895 = 1 2Thus ( 120601 1 2)

isin 1198711

(Ω) times 1198671



10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884


10038171003817100381710038171003817119865 (119875120576119896

(120601120576119896

1205951

120576119896

1205952

120576119896

)) minus 11991012057510038171003817100381710038171003817

2

119884

119877 ( 120601 1

2

)


(1205731

10038161003816100381610038161003816119867120576119896

(120601120576119896

)10038161003816100381610038161003816119861119881+ 1205732

10038171003817100381710038171003817120601120576119896

minus 1206010

10038171003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

120576119896

minus 120595119895

0

10038161003816100381610038161003816119861119881)

(55)

Therefore

G120572( 120601

1

2

)

=10038171003817100381710038171003817119865 (119902 (

1

2

)) minus 11991012057510038171003817100381710038171003817

2

119884

+ 120572119877 ( 120601 1

2

)


G120576119896120572(120601120576119896

1205951

120576119896

1205952

120576119896

)


G120576119896120572(120601119896 1205951

119896 1205952

119896)


10038171003817100381710038171003817119865 (119875120576119896

(120601119896 1205951

119896 1205952

119896)) minus 119910

12057510038171003817100381710038171003817

2

119884


(1205731

10038161003816100381610038161003816119867120576119896

(120601119896)10038161003816100381610038161003816119861119881+ 1205732

1003817100381710038171003817120601119896 minus 12060101003817100381710038171003817

2

1198671(Ω)

+ 1205733

2

sum

119895=1

10038161003816100381610038161003816120595119895

119896minus 120595119895

0

10038161003816100381610038161003816119861119881)

=10038171003817100381710038171003817119865 (119902 (119911

120572 1205951

120572 1205952

120572)) minus 119910

12057510038171003817100381710038171003817

2

119884

+ 120572119877 (119911120572 120601120572 1205951

120572 1205952

120572)

= G120572(119911120572 1206011

120572 1205951

120572 1205952

120572)

(56)

characterizing ( 120601 1205951120572 1205952


120572


120572 Since




120576120572in (50) with



Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Since 1198671015840


120576(119905) =

1120576 119905 isin (minus1205760)



120572 (Δ minus 119868) (120601 minus 1206010) = 119871120576120572120573

(120601 1205951

1205952

) in Ω (57)

(120601 minus 1206010) sdot 120584 = 0 at 120597Ω (58)

120572nabla sdot [

[

nabla (120595119895

minus 120595119895

0)

10038161003816100381610038161003816nabla (120595119895 minus 120595

119895

0)10038161003816100381610038161003816

]

]

= 119871119895

120576120572120573(120601 1205951

1205952

) 119895 = 1 2 (59)


119871120576120572120573

(120601 1205951

1205952

) = (1205951

minus 1205952

) 120573minus1

21198671015840

120576(120601)lowast

1198651015840

times (119875120576(120601 1205951

1205952

))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

minus 1205731(21205732)minus1

1198671015840

120576(120601) nabla sdot [

nabla119867120576(120601)

1003816100381610038161003816nabla119867120576 (120601)1003816100381610038161003816

]

(60)

1198711

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

(1198651015840

(119875120576(120601 1205951

1205952

))119867120576(120601))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(61)

1198712

120576120572120573(120601 1205951

1205952

) = (21205733)minus1

times (1198651015840

(119875120576(120601 1205951

1205952

)) (1 minus 119867120576(120601)))lowast

times (119865 (119875120576(120601 1205951

1205952

)) minus 119910120575

)

(62)


120576(120601)| and |nabla(120595119895 minus 120595119895

0)|








1205741119908 + 1205742119908120584= 119892 on 120597Ω

(63)







1

0(Ω) [40]


1198651 1198712

(Ω) 997888rarr 1198712

(120597Ω)

119906 997891997888rarr 1198651(119906) = 119908

120584

1003816100381610038161003816120597Ω

(64)



1(119906) = 119910


2






the assumption (A2)


1

(Ω) rarr 1198712



1198671(Ω)

le 11988811199061198712(Ω)

Let 119906119899 1199060isin 119863(119865) and 119908

119899 1199080




10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10038171003817100381710038171198651(119906119899) minus 1198651(1199060)10038171003817100381710038171198712(120597Ω)

le 1198621003817100381710038171003817119908119899 minus 1199080

10038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

10038171003817100381710038171198712(Ω)

le 1198621

1003817100381710038171003817119906119899 minus 119906010038171003817100381710038171198711(Ω)

(65)





120576120572


1= 1 and 120574





119896 1205952

119896)) minus 119910

120575

=

(119908119896)120584|120597Ωminus 119910120575 where 119908

119896solves


119896 1205952

119896) in Ω

119908119896= 119892 at 120597Ω

(66)

(2) evaluate ℎ119896= 1198651015840

1(119875120576(120601119896 1205951

119896 1205952

119896))lowast

(119903119896) isin 1198712

(Ω) solv-ing

Δℎ119896= 0 in Ω

ℎ119896= 119903119896 at 120597Ω

(67)

(3) calculate 120575120601119896= 119871120576120572120573(120601119896 1205951

119896 1205952

119896) and 120575120595119895

119896= 119871119895

120576120572120573sdot

(120601119896 1205951

119896 1205952

119896) as in (60) (61) and (62)


120595119895

119896 119895 = 1 2

120601119896+1= 120601119896+1

120572120575120601119896

120595119895

119896+1= 120595119895

119896+1

120572120575120595119895

119896

(68)


119896isin 1198712

(120597Ω) of the iterate(120601119896 120595119895




119896for the






119906 = 1205951 119909 isin 119863

1205952 119909 isin Ω119863

(69)




119908 = 119892 on 120597Ω(70)


0(Ω)




120584|120597Ω



1198652 119863 (119865) sub 119871

1

(Ω) 997888rarr 11986712

(120597Ω)

119906 997891997888rarr 1198652(119906) = 119908|

120597Ω

(71)


(Ω) is given



1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1= 1205831and

1205952= 1205832(|nabla119908|2


and 1205832= 1205832(|nabla119908|2


Remark 21 Note that 1198652(119906) = 119879

119863(119908) where 119879



(Ω) rarr 11986712

(120597Ω)


le

1198881199081198671(Ω)




1

(Ω) rarr

11986712






1198671



0nabla1199080) = 0 (72)


intΩ



1

0(Ω)

(73)




1003817100381710038171003817

2

1198712(Ω)le intΩ

119906119899


2

119889119909

le intΩ

1003816100381610038161003816(119906119899 minus 1199060)1003817100381710038171003817nabla1199080


(74)


0isin 1198821119901




1003817100381710038171003817

2

1198712(Ω)le1003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω)

times1003817100381710038171003817nabla1199080

1003817100381710038171003817119871119901(Ω)


(75)


1003817100381710038171003817119908119899 minus 119908010038171003817100381710038171198671(Ω)

le 1198621003817100381710038171003817119906119899 minus 1199060

1003817100381710038171003817119871119902(Ω) (76)




120576120572 Each


119899 1205951

119899 1205952



1


119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as in (60) (61) and


119899isin 1198671





119899 1205952

119899)) minus 119910

120575

=


nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119908) = 119891 in Ω

119908 = 119892 at 120597Ω(77)

(2) Evaluate 11986510158402(119875120576(120601119899 1205951

119899 1205952

119899))lowast



(Ω) solves

nabla sdot (119875120576(120601119899 1205951

119899 1205952

119899) nabla119907) = 0 in Ω

119907120584= 119903 at 120597Ω

(78)

(3) Calculate 119871120576120572120573(120601119899 1205951

119899 1205952

119899) and 119871119895

120576120572120573(120601119899 1205951

119899 1205952

119899) as



119899+1= 120601119899+ (1120572)120575120601


119899+ (1120572)120575120595

119895





120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120576120572


Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article On a Level-Set Method for Ill-Posed...

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