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intyre ed.)7 - Homage to Gaetano Fichera (A. Cialdea ed.)8 - Topics in Infinite Groups (M. Curzio and F. de Giovanni eds.)9 - Selected Topics in Cauchy-Riemann Geometry (S. Dragomir ed.)

10 - Topics in Mathematical Fluids Mechanics (G.P. Galdi and R. Rannacher eds.)11 - Model Theory and Applications (L. Belair, Z. Chatzidakis et al. eds.)12 - Topics in Diagram Geometry (A. Pasini ed.)13 - Complexity of Computations and Proofs (J. Krajíček ed.)14 - Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi (D. Pal-

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eds.)16 - Kinetic Methods for Nonconservative and Reacting Systems (G. Toscani ed.)17 - Set Theory: Recent Trends and Applications (A. Andretta ed.)18 - Selection Principles and Covering Properties in Topology (Lj.D.R. Kočinac ed.)20 - Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior

(M. Šilhavý ed.)21 - Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent

Developments (G. Casnati, F. Catanese, and R. Notari eds.)

22 - Theory and Applications of Proximity, Nearness and Uniformity (G. Di Maio andS. Naimpally eds.)

23 - On the notions of solution to nonlinear elliptic problems: results and developments(A. Alvino, A. Mercaldo, F. Murat, and I. Peral eds.)

Next issues

Trends in Incidence and Galois Geometries: a Tribute to Giuseppe Tallini (F. Maz-zocca, N. Melone and D. Olanda eds.)Numerical Methods for Balance Laws (G. Puppo and G. Russo eds.)

On the notions of solutionto nonlinear elliptic problems:results and developments

edited by Angelo Alvino, Anna Mercaldo,François Murat, and Ireneo Peral

ISBN 978-88-548-3032-5

Receive December, 2009

c⃝ 2008 by Dipartimento di Matematica della Seconda Università di Napoli

Photocomposed copy prepared from a LATEX file.

Editors’ addresses:

Angelo AlvinoDipartimento di Matematicae Applicazioni “R. Caccioppoli”Università degli Studi di Napoli Federico IIVia Cintia, Monte S. AngeloI-80126 Napoli, Italyemail: [email protected]

François MuratLaboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris VI)Boîte courrier 187

-75252 Paris cedex 05, Franceemail: [email protected]

Anna MercaldoDipartimento di Matematicae Applicazioni “R. Caccioppoli”Università degli Studi di Napoli Federico IIVia Cintia, Monte S. AngeloI-80126 Napoli, Italyemail: [email protected]

Ireneo PeralDepartamento de MatemáticasUniversidad Autónoma de MadridCampus de Cantoblanco

28049 Madrid, Spainemail: [email protected]

F CinMoro,

Authors’ addresses:

Waad Al SayedLaboratoire de Mathématiqueset Physique ThéoriqueCNRS UMR 6083Université François RabelaisF-37200 Tours, Franceemail: [email protected]

Isabeau BirindelliDipartimento di Matematica“G. Castelnuovo”“Sapienza” – Università di RomaPiazzale Aldo Moro, 2I-00185 Roma, Italyemail: [email protected]

Lucio BoccardoDipartimento di Matematica“G. Castelnuovo”“Sapienza” – Università di RomaPiazzale Aldo Moro, 2I-00185 Roma, Italyemail: [email protected]

Françoise DemengelDépartement de MathématiquesUniversité de Cergy-PontoiseSite de Saint Martin2, avenue Adolphe ChauvinF-95302 Cergy-Pontoise Cedex, Franceemail: [email protected]

Nicolas ForcadelCEREMADE, UMR CNRS 7534Université Paris-DauphinePlace de Lattre de TassignyF-75775 Paris Cedex 16, Franceemail: [email protected]

Kari AstalaDepartment of Mathematics and StatisticsUniversity of HelsinkiFI-00014 Helsinki, Finlandemail: [email protected]

Dominique BlanchardLaboratoire de MathématiquesRaphaël SalemUniversité de Rouen et CNRSAvenue de l’Université, BP12F-76801 Saint-Étienne du Rouvray, Franceemail: [email protected]

Vicent CasellesDepartament de TecnologiaUniversitat Pompeu-FabraPasseig de Circumvalacio, 808003 Barcelona, Spainemail: [email protected]

Patricio FelmerDepartamento de Ingenieria Matematicay Centro de Modelamiento MatematicoUMR2071 CNRS-UChileUniversidad de ChileCasilla 170 Correo 3, Santiago, Chileemail: [email protected]

Olivier GuibéLaboratoire de MathématiquesRaphaël SalemUniversité de Rouen et CNRSAvenue de l’Université, BP12F-76801 Saint-Étienne du Rouvray, Franceemail: [email protected]

Moro,

Noureddine IgbidaLAMFA, CNRS UMR 6140Université de Picardie Jules Verne33, rue Saint LeuF-80039 Amiens Cedex 1, Franceemail: [email protected]

Tadeusz IwaniecDepartment of MathematicsSyracuse University215 Carnegie HallSyracuse, NY 13244-1150, U.S.A.email: [email protected]

Andrea MalchiodiSector of Mathematical AnalysisSISSA, via Beirut 2-4I-34014 Trieste, Italyemail: [email protected]

José M. MazónDepartament d’Anàlisi MatemàticaUniversitat de ValènciaC/ Dr. Moliner, 5046100 Burjassot (València), Spainemail: [email protected]

Régis MonneauUniversité Paris-EstCermics, Ecole des Ponts ParisTech6-8 avenue Blaise PascalF-77455 Marne la Vallee Cedex 2, Franceemail: [email protected]

Cyril ImbertCEREMADE, UMR CNRS 7534Université Paris-DauphinePlace de Lattre de TassignyF-75775 Paris Cedex 16, Franceemail: [email protected]

Alexander A. KovalevskyInstitute of Applied Mathematicsand MechanicsNational Academy of Sciences of UkraineR. Luxemburg St. 7483114 Donetsk, Ukraineemail: [email protected]

Gaven J. MartinInstitute for Advanced StudyInstitute of Information andMathematical SciencesMassey UniversityAlbany, Auckland, New Zealandemail: [email protected]

Giuseppe MingioneDipartimento di MatematicaUniversità di ParmaViale G. P. Usberti 53/a, CampusI-43100 Parma, Italyemail: [email protected]

Luigi OrsinaDipartimento di Matematica“G. Castelnuovo”“Sapienza” – Università di RomaPiazzale Aldo Moro, 2I-00185 Roma, Italyemail: [email protected]

Alessio PorrettaDipartimento di MatematicaUniversitá di Roma “Tor Vergata”Via della Ricerca Scientifica, 1I-00133 Roma, Italyemail: [email protected]

José ToledoDepartament d’Anàlisi MatemàticaUniversitat de ValènciaC/ Dr. Moliner, 5046100 Burjassot (València), Spainemail: [email protected]

Alexander QuaasDepartamento de MatematicaUniversidad Santa MariaCasilla: V-110, AvdaEspana 1680, Valparaiso, Chileemail: [email protected]

Laurent VéronLaboratoire de Mathématiqueset Physique ThéoriqueCNRS UMR 6083Université François RabelaisF-37200 Tours, Franceemail: [email protected]

Contents

Preface

Solutions of Some Nonlinear Parabolic Equations with Initial Blow-up 1Waad Al Sayed and Laurent Véron

Some Flux-Limited Quasi-Linear Elliptic Equations without Coercivity 25Fuensanta Andreu, Vicent Caselles, and José M. Mazón

Degenerate Elliptic Equations with Nonlinear Boundary Conditions 67Fuensanta Andreu, Noureddine Igbida, José M. Mazón, and José Toledo

Bi-Lipschitz Homeomorphisms of the Circle and Non-LinearBeltrami Equations 105Kari Astala, Tadeusz Iwaniec, and Gaven J. Martin

Bifurcation for Singular Fully Nonlinear Equations 117Isabeau Birindelli and Françoise Demengel

A few Results on Coupled Systems of Thermomechanics 145Dominique Blanchard

Nonlinear Elliptic Problems with Singular and Natural GrowthLower Order Terms 183Lucio Boccardo and Luigi Orsina

Around Viscosity Solutions for a Class of Superlinear SecondOrder Elliptic Differential Equations 205Patricio Felmer and Alexander Quaas

Viscosity Solutions for Particle Systems and Homogenizationof Dislocation Dynamics 229Nicolas Forcadel, Cyril Imbert, and Régis Monneau

Uniqueness of the Renormalized Solution to a Class ofNonlinear Elliptic Equations 255Olivier Guibé

Nonlinear Fourth-Order Equations with a StrengthenedEllipticity and L1-data 283Alexander A. Kovalevsky

On a Class of Nonlinear Equations with Exponential Nonlinearitiesand Measure Data 339Andrea Malchiodi

Towards a Non-Linear Calderón-Zygmund Theory 371Giuseppe Mingione

On the Comparison Principle for p-Laplace Type Operatorswith First Order Terms 459Alessio Porretta

Solutions of Some NonlinearParabolic Equationswith Initial Blow-up

Waad Al Sayed and Laurent Véron

Contents

1. Introduction (3).2. Minimal and maximal solutions (4).3. Uniqueness of large solutions (18).

Degenerate Elliptic Equations withNonlinear Boundary Conditions

F. Andreu, N. Igbida, J.M. Mazón, and J. Toledo

Contents

1. Introduction (3).2. Preliminaries (9).3. Integrable data (14).4. An obstacle problem (20).5. Measure data (26).6. Applications (33).

Degenerate Elliptic Equations with Nonlinear Boundary Conditions 3

1. Introduction

The purpose of this survey is to present some recent results given by theauthors about existence and uniqueness of solutions for a degenerate ellipticproblem with nonlinear boundary condition of the form

(Sγ,βµ1,µ2)

−div a(x,Du) + γ(u) 3 µ1 in Ω

a(x,Du) · η + β(u) 3 µ2 on ∂Ω,

where Ω is a bounded domain in RN with smooth boundary ∂Ω, the functiona : Ω×RN → RN is a Carathéodory function satisfying the classical Leray-Lionsconditions, η is the unit outward normal on ∂Ω, µ1 = µ1 ∂Ω, µ2 = µ2 Ωare measures and γ and β are maximal monotone graphs in R2 (see, e.g.,[23]), 0 ∈ γ(0) ∩ β(0). General nonlinear diffusion operators of Leray-Lionstype, different from the Laplacian, appear when one deals with non-Newtonianfluids (see, e.g., [7]).

The nonlinearities γ and β satisfy rather general assumptions. In particu-lar, they may be multivalued and this allows to include the Dirichlet condition(taking β to be the monotone graph D defined by D(0) = R) and the non ho-mogeneous Neumann boundary condition (taking β to be the monotone graphN defined by N(r) = 0 for all r ∈ R) as well as many other nonlinear fluxes onthe boundary that occur in some problems in Mechanic and Physics (see, e.g.,[34] or [22]). For instance, in the Signorini problem (see, e.g., [36], [37], [28])which appears in elasticity and corresponds to the monotone graph

β(r) =

∅ if r < 0]−∞, 0] if r = 00 if r > 0,

in problems of optimal control of temperature and in the modelling of semiper-meability (see [34]), which corresponds in some cases to the monotone graph

β(r) =

∅ if r < a

]−∞, 0] if r = a

0 if r ∈]a, b[[0,+∞[ if r = b

∅ if r > b,

4 F. Andreu, N. Igbida, J.M. Mazón, and J. Toledo

where a < 0 < b.Note also that, since γ may be multivalued, problems of type (Sγ,βµ1,µ2

) ap-pears in various phenomena with changes of state like multiphase Stefan prob-lem (cf. [30]) and in the weak formulation of the mathematical model of the socalled Hele Shaw problem (cf. [32] and [35]). In the case in which D(γ) 6= R weare dealing with obstacle problems, also called unilateral problems in the lit-erature. Obstacle problems appear in different physical context, for instance,in deformation of membrane constrained by an obstacle, in bending of elas-tic isotropic homogeneous plat over an obstacle and in cavitation problemsin hydrodynamic lubrication. Notice also that some free boundary problemsfall into this scope by using Baiocchi transformation (see [8]), for more detailsconcerning physical applications we refer to [44] or [34].

In the particular case a(x, ξ) = ξ, the problem (Sγ,βµ1,µ2) reads

(Lγ,βµ1,µ2)

−∆u+ γ(u) 3 µ1 in Ω

∂ηu+ β(u) 3 µ2 on ∂Ω,

where ∂ηu simply denotes the outward normal derivative of u. For this kind ofproblems in the homogeneous case, µ2 ≡ 0, the pioneering works are the paperby H. Brezis ([22]), in which problem (Lγ,βµ1,0

) is studied for γ the identity, βa maximal monotone graph and µ1 ∈ L2(Ω), and the paper by H. Brezis andW.Strauss ([27]), in which problem (Lγ,βµ1,0

) is studied for µ1 ∈ L1(Ω) and γ,β continuous nondecreasing functions from R into R with γ′ ≥ ε > 0. Theseworks were extended by Ph. Bénilan, M. G. Crandall and P. Sacks in [17]where they study problem (Sγ,βµ1,0

) for any γ and β maximal monotone graphsin R2 such that 0 ∈ γ(0) and 0 ∈ β(0), and prove, between other results, that,for any µ1 ∈ L1(Ω) satisfying the range condition

infRan(γ)meas(Ω) + infRan(β)meas(∂Ω) <∫

Ω

µ1

< supRan(γ)meas(Ω) + supRan(β)meas(∂Ω),

there exists a unique, up to a constant for u, named weak solution, [u, z, w] ∈W 1,1(Ω) × L1(Ω) × L1(∂Ω), z(x) ∈ γ(u(x)) a.e. in Ω, w(x) ∈ β(u(x)) a.e. in


∂Ω, such that ∫Ω

Du ·Dv +∫

Ω

zv +∫∂Ω

wv =∫

Ω

µ1v,

for all v ∈W 1,∞(Ω).For p-Laplacian type equations, an important work in the L1-theory is [12],

where problem

(Dγφ)

−div a(x,Du) + γ(u) 3 µ1 in Ω

u = 0 on ∂Ω

is studied for any γ maximal monotone graph in R2 such that 0 ∈ γ(0). It isproved that, for any µ1 ∈ L1(Ω), there exists a unique, named entropy solution,[u, z] ∈ T 1,p

0 (Ω)× L1(Ω), z(x) ∈ γ(u(x)) a.e. in Ω, such that∫Ω

a(., Du) ·DTk(u− v) +∫

Ω

zTk(u− v) ≤∫

Ω

µ1Tk(u− v) ∀k > 0,

for all v ∈ L∞(Ω) ∩W 1,p0 (Ω) (see Section 2 for the definition of T 1,p

0 (Ω)).In [2], [4] and [5] the results of [17] and [12] are extended by proving the

existence and uniqueness of weak, entropy, renormalized or generalized weaksolutions for the general non homogeneous problem (Sγ,βµ1,µ2

) depending on thedata µ1, µ2. The arguments of the proofs are very connected to the nature ofthe nonlinearities γ and β. Grosso modo the following cases are studied,

(A) D(γ) = R, either D(β) = R or div a(x,Du) = ∆p(u), and µ1, µ2 inte-grable data;µ2 ≡ 0, either D(β) = R or div a(x,Du) = ∆p(u), and µ1 integrable data(no conditions on γ);

(B) R 6= D(γ) ⊂ D(β) and µ1, µ2 integrable data (an obstacle problem);

(C) D(γ) = D(β) = R and µ1 + µ2 a diffuse measure, that is, it does notcharge sets of zero p−capacity.

The main interest in this study is that we are dealing with general nonlinearoperators −div a(x,Du) with nonhomogeneous boundary conditions, which is


quite different from the homogeneous case µ2 = 0, and general nonlinearitiesγ and β. As in [17], a range condition relating the average of µ1 and µ2 tothe range of β and γ is necessary for existence of weak solution and entropysolution (see Remarks 3.1 and 5.2).

However, in contrast to the smooth homogeneous case µ2 = 0, even fora corresponding to the Laplacian, for the nonhomogeneous case this rangecondition is not sufficient for the existence of weak solution. The intersectionof the domains of β and γ creates an obstruction phenomena for the existenceof these solutions. Even if D(β) = R it does not exist weak solution as thefollowing example shows. Let γ be such that D(γ) = [0, 1], β = R × 0,and let µ1 ∈ L1(Ω), µ2 ≤ 0 a.e. in Ω, and µ2 ∈ L1(∂Ω), µ2 ≤ 0 a.e. in ∂Ω.If there exists [u, z, w] weak solution of problem (Lγ,0µ1,µ2

) (see Definition 3.1),then z ∈ γ(u), therefore 0 ≤ u ≤ 1 a.e. in Ω, w = 0, and it holds that for anyv ∈W 1,p(Ω) ∩ L∞(Ω),∫

Ω

a(x,Du)Dv +∫

Ω

zv =∫

Ω

µ1v +∫∂Ω

µ2v.

Taking v = u, as u ≥ 0, we get

0 ≤∫

Ω

a(x,Du)Du+∫

Ω

zu =∫

Ω

µ1u+∫∂Ω

µ2u ≤ 0.

Therefore, we obtain that∫

Ω|Du|p = 0, so u is constant and∫

Ω

zv =∫

Ω

µ2v +∫∂Ω

µ1v,

for any v ∈W 1,p(Ω)∩L∞(Ω), and in particular, for any v ∈W 1,p0 (Ω)∩L∞(Ω).

Consequently, µ1 = z a.e. in Ω, and µ2 must be 0 a.e. in ∂Ω.

In general, for obstacle problems the existence of weak solution, in theusual sense, fails to be true for nonhomogeneous boundary conditions, so anew concept of solution has to be introduced.

For the case where the data are Radon measures, the problem is againdifferent. There is a large literature on elliptic problems with measure data,mainly for the homogeneous Dirichlet problem and γ ≡ 0, that is, for the


problem

(S0,Dµ,0 )

−div a(x,Du) = µ in Ω

u = 0 on ∂Ω.

In the linear case, existence and uniqueness of solutions of (S0,Dµ,0 ) was obtained

by G. Stampacchia [45] by duality techniques. In the nonlinear case the firstattempt to solve problem (S0,D

µ,0 ) was done by L. Boccardo and T. Gallouët,who proved in [18] and [19] the existence of weak solutions of (S0,D

µ,0 ) under theassumption p > 2 − 1

N . In the case 1 < p ≤ 2 − 1N , even for the particular

case µ ∈ L1(Ω), the definition of weak solution is not enough in order to getuniqueness. It was necessary to find some extra conditions on the distributionalsolutions of (S0,D

µ,0 ) in order to ensure both existence and uniqueness. This wasdone by Ph. Bénilan et alt. for the case of measures in L1(Ω), by introducingthe concept of entropy solution in [12], and by P. L. Lions and F. Murat in anunpublished paper where the concept of renormalized solution was introduced.For diffuse measures, that is, for measures in L1(Ω) +W−1,p′

(Ω), the problemwas solved by L. Boccardo, T. Gallouët and L. Orsina in [20], and for generalmeasures by G. Dal Maso et alt. in [31].

The study of the homogeneous Dirichlet problem for the Laplacian andγ 6≡ 0 was initiated by Ph. Bénilan and H. Brezis in 1975 (see [13]) for theparticular case γ(r) = gp(r) := |r|p−1r. They proved the existence of weaksolutions of problem

(Lγ,Dµ,0 )

−∆u+ γ(u) = µ in Ω

u = 0 on ∂Ω,

for any measure µ if p < NN−2 (N ≥ 2), and non existence if p ≥ N

N−2 (N ≥ 3)for µ = δa, with a ∈ Ω. Problem (Lγ,Dµ,0 ) was also studied by P. Baras andM. Pierre [9]. Recently it has been studied by H. Brezis, M. Marcus andA. C. Ponce in [25] in the case of a continuous nondecreasing nonlinearityγ : R → R, γ(0) = 0 (see also [46], [10] for the particular case γ(r) = er − 1).The same problem has been studied by H. Brezis and A. C. Ponce [26] in thecase Dom(γ) 6= R closed. The case Dom(γ) 6= R open has been studied by L.Dupaigne, A. C. Ponce and A. Porreta [33].


The study of nonlinear equations involving measures as boundary conditionwas initiated by A. Gmira and L. Veron [38]. They proved the existence of weaksolutions of problem

(GV )

−∆u+ |u|q−1u = 0 in Ω

u = µ on ∂Ω,

for any Radon measure µ on ∂Ω in the subcritical case 1 < q < N+1N−1 . In the

supercritical case, q ≥ N+1N−1 , this is no longer true; for instance, the problem has

no solution if the measure µ is concentrated at a single point. M. Marcus andL. Veron in [42] characterized the Radon measures µ on ∂Ω for which problem(GV ) has solution in the supercritical case, these measures are those that areabsolutely continuous respect to the Bessel capacity C 2

q ,q′ on ∂Ω. In the last

years an extensive study of removable singularities and boundary traces for thistype of problems has been done by M. Marcus and L. Veron (see [43] and thereferences therein).

The study of reduced measures initiated in [25] by H. Brezis, M. Marcusand A. C. Ponce for problem (Lγ,Dµ,0 ) has been developed in [26] by H. Brezisand A. C. Ponce for problems of the form

(BP )

−∆u+ γ(u) = 0 in Ω

u = µ on ∂Ω,

where γ : R → R is a nondecreasing continuous function with γ(r) = 0 forall r ≤ 0. In that paper the authors make the observation that in all theabove problems the equation in Ω is nonlinear but the boundary conditionsis the usual Dirichlet boundary condition, being also interesting to investigateproblems with nonlinear boundary conditions of type

(Lg1,β0,µ )

−∆u+ u = 0 in Ω

∂u

∂η+ β(u) 3 µ on ∂Ω,

where β is a maximal monotone graph in R2.


2. Preliminaries

Throughout this article, Ω ⊂ R is a bounded domain with boundary ∂Ωof class C1, p > 1, γ and β are maximal monotone graphs in R2 such that0 ∈ γ(0) ∩ β(0) and a : Ω× RN → RN is a Carathéodory function such that

(H1) there exists Λ > 0 such that a(x, ξ) · ξ ≥ Λ|ξ|p for a.e. x ∈ Ω and for allξ ∈ RN ,

(H2) there exists σ > 0 and % ∈ Lp′(Ω) such that |a(x, ξ)| ≤ σ(%(x) + |ξ|p−1)

for a.e. x ∈ Ω and for all ξ ∈ RN , where p′ = pp−1 ,

(H3) (a(x, ξ1) − a(x, ξ2)) · (ξ1 − ξ2) > 0 for a.e. x ∈ Ω and for all ξ1, ξ2 ∈RN , ξ1 6= ξ2.

The hypotheses (H1−H3) are classical in the study of nonlinear operators indivergence form (cf., [41]). The model example of a function a satisfying thesehypotheses is a(x, ξ) = |ξ|p−2ξ. The corresponding operator is the p-Laplacianoperator ∆p(u) = div(|Du|p−2Du).

We denote by LN the N−dimensional Lebesgue measure of RN and byHN−1 the (N − 1)-dimensional Hausdorff measure.

For 1 ≤ p < +∞, Lp(Ω) and W 1,p(Ω) denote respectively the standardLebesgue space and Sobolev space, and W 1,p

0 (Ω) is the closure of D(Ω) inW 1,p(Ω). For u ∈ W 1,p(Ω), we denote by u or τ(u) the trace of u on ∂Ω inthe usual sense and by W

1p′ ,p(∂Ω) the set τ(W 1,p(Ω)). Recall that Ker(τ) =

W 1,p0 (Ω).We write

sign0(r) :=

1 if r > 0,0 if r = 0,−1 if r < 0,

sign+0 (r) :=

1 if r > 0,0. if r ≤ 0,

and for k > 0,Tk(s) = sup(−k, inf(s, k)).

In [12], the authors introduce the set

T 1,p(Ω) = u : Ω −→ R measurable such that Tk(u) ∈W 1,p(Ω) ∀k > 0.


They also prove that given u ∈ T 1,p(Ω), there exists a unique measurablefunction v : Ω→ RN such that

DTk(u) = vχ|v|<k ∀k > 0.

This function v will be denoted by Du. It is clear that if u ∈ W 1,p(Ω), thenv ∈ Lp(Ω) and v = Du in the usual sense.

As in [6], T 1,ptr (Ω) denotes the set of functions u in T 1,p(Ω) satisfying the

following conditions, there exists a sequence un in W 1,p(Ω) such that

(a) un converges to u a.e. in Ω,

(b) DTk(un) converges to DTk(u) in L1(Ω) for all k > 0,

(c) there exists a measurable function v on ∂Ω, such that un converges to va.e. in ∂Ω.

The function v is the trace of u in the generalized sense introduced in [6]. Inthe sequel, the trace of u ∈ T 1,p

tr (Ω) on ∂Ω will be denoted by tr(u) or u. Letus recall that in the case where u ∈W 1,p(Ω), tr(u) coincides with the trace ofu, τ(u), in the usual sense, and the space T 1,p

0 (Ω), introduced in [12] to study(Dγ

φ), is equal to Ker(tr). Moreover, for every u ∈ T 1,ptr (Ω) and every k > 0,

τ(Tk(u)) = Tk(tr(u)), and, if φ ∈W 1,p(Ω)∩L∞(Ω), then u− φ ∈ T 1,ptr (Ω) and

tr(u− φ) = tr(u)− τ(φ).We denote

V 1,p(Ω) :=φ ∈ L1(Ω) : ∃M > 0 such that∫

Ω

|φv| ≤M‖v‖W 1,p(Ω) ∀v ∈W 1,p(Ω)

and

V 1,p(∂Ω) :=ψ ∈ L1(∂Ω) : ∃M > 0 such that∫∂Ω

|ψv| ≤M‖v‖W 1,p(Ω) ∀v ∈W 1,p(Ω).

V 1,p(Ω) is a Banach space endowed with the norm

‖φ‖V 1,p(Ω) := infM > 0 :

∫Ω

|φv| ≤M‖v‖W 1,p(Ω) ∀v ∈W 1,p(Ω),


and V 1,p(∂Ω) is a Banach space endowed with the norm

‖ψ‖V 1,p(∂Ω) := infM > 0 :

∫∂Ω

|ψv| ≤M‖v‖W 1,p(Ω) ∀v ∈W 1,p(Ω).

Observe that, Sobolev embeddings and Trace theorems imply, for 1 ≤ p < N ,

Lp′(Ω) ⊂ L(Np/(N−p))′(Ω) ⊂ V 1,p(Ω)

andLp

′(∂Ω) ⊂ L((N−1)p/(N−p))′(∂Ω) ⊂ V 1,p(∂Ω).

Also,V 1,p(Ω) = L1(Ω) and V 1,p(∂Ω) = L1(∂Ω) when p > N,

Lq(Ω) ⊂ V 1,N (Ω) and Lq(∂Ω) ⊂ V 1,N (∂Ω) for any q > 1.

For an open bounded set U of RN , the p-capacity relative to U , Cp(., U), isdefined in the following way. For any compact subset K of U ,

Cp(K,U) = inf∫

U

|Du|p ; u ∈ C∞c (U), u ≥ χK,

where χK is the characteristic function of K; we will use the convention thatinf ∅ = +∞. The p-capacity of any open subset O ⊂ U is defined by

Cp(O,U) = sup Cp(K) ; K ⊂ O compact .

Finally, the p-capacity of any Borel set A ⊂ U is defined by

Cp(A,U) = inf Cp(O) ; O ⊂ A open .

A function u defined on U is said to be capp-quasi-continuous in A ⊂ U if forevery ε > 0, there exists an open set Bε ⊆ U with Cp(Bε, U) < ε such that therestriction of u to A \Bε is continuous. It is well known that every function inW 1,p(U) has a capp-quasi-continuous representative, whose values are definedcapp-quasi everywhere in U , that is, up to a subset of U of zero p-capacity.When we are dealing with the pointwise values of a function u ∈ W 1,p(U), wealways identify u with its capp-quasi-continuous representative.

Let us remark that if u ∈ T 1,p(Ω), then u has a capp-quasi-continuousrepresentative, which will be denoted equally by u; the capp-quasi-continuous


representative can be infinite on a set of positive p-capacity (see [31]). If inaddition the function u ∈ T 1,p(Ω) is assumed to satisfy the estimate∫

Ω

|DTk(u)|p dx ≤ C(k + 1) ∀ k > 0,

where C is independent of k, then the capp-quasi-continuous representative ofu is capp-quasi every where finite (see [31]).

Since we are considering Ω to be a bounded domain in RN with ∂Ω ofclass C1, Ω is an extension domain (see [24]), so we can fix an open boundedsubset UΩ of RN such that Ω ⊂ UΩ, and there exists a bounded linear operatorE : W 1,p(Ω)→W 1,p

0 (UΩ) for which

(i) E(u) = u a.e in Ω for each u ∈W 1,p(Ω),

(ii) ‖E(u)‖W 1,p0 (UΩ) ≤ C‖u‖W 1,p(Ω), where C is a constant depending only on

p and Ω.

We call E(u) an extension of u to UΩ. If u ∈ W 1,p(Ω), 1 < p ≤ ∞, itis possible to give a pointwise definition of the trace τ(u) of u on ∂Ω in thefollowing way (see [47]), as E(u) ∈W 1,p

0 (UΩ), every point of UΩ, except possiblya set of zero p−capacity, is a Lebesgue point of E(u). Since p > 1, the setsof zero p−capacity are of HN−1-measure zero and therefore E(u) is definedHN−1-almost everywhere on ∂Ω, so τ(u) = E(u) on ∂Ω. This definition isindependent of the open set UΩ and also of the extension E(u). From now onUΩ will be a fix open bounded subset of RN such that Ω ⊂ UΩ. We denoteτ(u) by u in the rest of the paper.

Given u ∈ T 1,p(Ω) there exists u ∈ T 1,p0 (UΩ) such that

Tk(u) = E[Tk(u)] for all k > 0.

For U an open subset of RN , we set by Mb(U) the space of all Radonmeasures in U with bounded total variation. We recall that for a measure µ ∈Mb(U) and a Borel set A ⊂ U , the measure µ A is defined by (µ A)(B) =µ(B ∩ A) for any Borel set B ⊂ U . If a measure µ ∈ Mb(U) is such thatµ = µ A for a certain Borel set A, the measure µ is said to be concentratedon A. For µ ∈Mb(U), we denote by µ+, µ− and |µ| the positive part, negative


part and the total variation of the measure µ, respectively. By µ = µa +µs wedenote the Radon-Nikodym decomposition of µ relatively to LN . For simplicity,we write also µa for its density respect to LN , that is, for the function f ∈ L1(U)such that µa = fLN U .

We denote by Mpb(U) the space of all diffuse Radon measures in U , i.e.,

measures which do not charge sets of zero p−capacity. In [20] it is proved thatµ ∈Mb(U) belongs toMp

b(U) if and only if it belongs to L1(U) +W−1,p′(U),

where W−1,p′(U) = [W 1,p

0 (U)]∗. Moreover, if u ∈ W 1,p(U) and µ ∈ Mpb(U),

then u is measurable with respect to µ. If u further belongs to L∞(U), then ubelongs to L∞(U, dµ), hence to L1(U, dµ).

Let ϑ be a maximal monotone graph in R × R. For r ∈ N, the Yosidaapproximation ϑr of ϑ is given by ϑr = r(I − (I + 1

rϑ)−1). The function ϑr ismaximal monotone and Lipschitz. We recall the definition of the main sectionϑ0 of ϑ

ϑ0(s) :=

the element of minimal absolute value of ϑ(s) if ϑ(s) 6= ∅,

+∞ if [s,+∞) ∩Dom(ϑ) = ∅,

−∞ if (−∞, s] ∩Dom(ϑ) = ∅.

We have that |ϑr| is increasing in r, if s ∈ Dom(ϑ), ϑr(s)→ ϑ0(s) as r → +∞,and if s /∈ Dom(θ), |ϑr(s)| → +∞ as r → +∞. If 0 ∈ Dom(ϑ), jϑ(r) =∫ r

0ϑ0(s)ds defines a convex lower semi-continuous function such that ϑ = ∂jϑ.

If j∗ϑ is the Legendre transformation of jϑ then ϑ−1 = ∂j∗ϑ.We set

ϑ(r+) := inf ϑ(]r,+∞[), ϑ(r−) := supϑ(]−∞, r[)

for r ∈ R, where we use the conventions inf ∅ = +∞ and sup ∅ = −∞. It iseasy to see that

ϑ(r) = [ϑ(r−), ϑ(r+)] ∩ R for r ∈ R.

Moreover,J(ϑ) := θ ∈ Dom(ϑ) : ϑ(r−) < ϑ(r+)


is a countable set.In [15] the following relation for u, v ∈ L1(Ω) is defined,

u v if∫Ω

(u− k)+ ≤∫

Ω

(v − k)+ and∫

Ω

(u+ k)− ≤∫

Ω

(v + k)− for any k > 0.

We finish this section with the following definition.

Definition 2.1 (([6]). We say that a is smooth when, for any φ ∈ L∞(Ω)such that there exists a bounded weak solution u of the homogeneous Dirichletproblem

(D)

− div a(x,Du) = φ in Ωu = 0 on ∂Ω,

there exists ψ ∈ L1(∂Ω) such that u is also a weak solution of the Neumannproblem

(N)

− div a(x,Du) = φ in Ωa(x,Du) · η = ψ on ∂Ω.

Functions a corresponding to linear operators with smooth coefficients andp-Laplacian type operators are smooth (see [22] and [40]). The smoothness ofthe Laplacian operator is even stronger than this, in fact, there is a boundedlinear mapping T : L1(Ω) → L1(∂Ω), such that the weak solution of (D) forφ ∈ L1(Ω) is also a weak solution of (N) for ψ = T (φ) (see [17]).

3. Integrable data

In this section we deal with integrable data, so we rewrite µ1 = φ andµ2 = ψ in order to denote functions. Let us begin by giving the differentconcepts of solutions we use.

Definition 3.1. Let φ ∈ L1(Ω) and ψ ∈ L1(∂Ω). A triple of functions[u, z, w] ∈ W 1,p(Ω) × L1(Ω) × L1(∂Ω) is a weak solution of problem (Sγ,βφ,ψ)if z(x) ∈ γ(u(x)) a.e. in Ω, w(x) ∈ β(u(x)) a.e. in ∂Ω, and∫

Ω

a(x,Du) ·Dv +∫

Ω

zv +∫∂Ω

wv =∫∂Ω

ψv +∫

Ω

φv,

for all v ∈ L∞(Ω) ∩W 1,p(Ω).


In general, as it is remarked in [12], for 1 < p ≤ 2− 1N , there exists f ∈ L1(Ω)

such that the problem

u ∈W 1,1loc (Ω), u−∆p(u) = f in D′(Ω),

has no solution. In [12], to overcome this difficulty and to get uniqueness, itwas introduced a new concept of solution, named entropy solution. Followingthese ideas, we introduce the following concept of solution.

Definition 3.2. Let φ ∈ L1(Ω) and ψ ∈ L1(∂Ω). A triple of functions[u, z, w] ∈ T 1,p

tr (Ω)× L1(Ω)× L1(∂Ω) is an entropy solution of problem (Sγ,βφ,ψ)if z(x) ∈ γ(u(x)) a.e. in Ω, w(x) ∈ β(u(x)) a.e. in ∂Ω and

(3.1)

∫Ω

a(x,Du) ·DTk(u− v) +∫

Ω

zTk(u− v) +∫∂Ω

wTk(u− v)

≤∫∂Ω

ψTk(u− v) +∫

Ω

φTk(u− v) ∀k > 0,


Obviously, every weak solution is an entropy solution and an entropy solu-tion with u ∈W 1,p(Ω) is a weak solution.

Remark 3.1 - If we take v = Th(u)± 1 as test function in (3.1) and let h go to+∞, we get that ∫

Ω

z +∫∂Ω

w =∫∂Ω

ψ +∫

Ω

φ.

Then necessarily φ and ψ must satisfy

R−γ,β ≤∫∂Ω

ψ +∫

Ω

φ ≤ R+γ,β ,

whereR+γ,β := supRan(γ)meas(Ω) + supRan(β)meas(∂Ω)

andR−γ,β := infRan(γ)meas(Ω) + infRan(β)meas(∂Ω).

In general, we will suppose R−γ,β < R+γ,β and write Rγ,β :=]R−γ,β ,R

+γ,β [.

The following result holds for entropy solutions.


Lemma 3.1. Let [u, z, w] be an entropy solution of problem (Sγ,βφ,ψ). Then, forall h > 0,

λ

∫h<|u|<h+k

|Du|p ≤ k∫∂Ω∩|u|≥h

|ψ|+ k

∫Ω∩|u|≥h

|φ|.

Respect to uniqueness we have the following results.

Theorem 3.1 (([2]). Let φ ∈ L1(Ω) and ψ ∈ L1(∂Ω), and let [u1, z1, w1]and [u2, z2, w2] be entropy solutions of problem (Sγ,βφ,ψ). Then, there exists aconstant c ∈ R such that

u1 − u2 = c a.e. in Ω,

z1 − z2 = 0 a.e. in Ω.

w1 − w2 = 0 a.e. in ∂Ω.

Moreover, if c 6= 0, there exists a constant k ∈ R such that z1 = z2 = k.

Since every weak solution is an entropy solution of problem (Sγ,βφ,ψ), the sameresult is true for weak solutions. Nevertheless we have the following generalresult for weak solutions, in fact, a contraction principle between sub and superweak solutions. As usual, we understand weak sub and supersolution in thefollowing way. A triple of functions [u, z, w] ∈ W 1,p(Ω)× L1(Ω)× L1(∂Ω) is aweak subsolution (resp. supersolution) of problem (Sγ,βφ,ψ) if z(x) ∈ γ(u(x)) a.e.in Ω, w(x) ∈ β(u(x)) a.e. on ∂Ω, and∫

Ω

a(x,Du) ·Dv +∫

Ω

zv +∫∂Ω

w v ≤ (resp. ≥)∫

Ω

φv +∫∂Ω

ψv,

for all v ∈ L∞(Ω) ∩W 1,p(Ω), v ≥ 0.

Theorem 3.2 (([4]). Let φ1, φ2 ∈ L1(Ω), ψ1, ψ2 ∈ L1(∂Ω). If [u1, z1, w1] is aweak subsolution of (Sγ,βφ1,ψ1

) and [u2, z2, w2] is a weak supersolution of (Sγ,βφ2,ψ2),

then ∫Ω

(z1 − z2)+ +∫∂Ω

(w1 − w2)+ ≤∫

Ω

(φ1 − φ2)+ +∫∂Ω

(ψ1 − ψ2)+.

Respect to the existence of weak solutions we obtain the following results.


Theorem 3.3 (([2]). Assume D(γ) = R, either D(β) = R or a smooth, andR−γ,β < R

+γ,β .

For any φ ∈ V 1,p(Ω) and ψ ∈ V 1,p(∂Ω) with∫Ω

φ+∫∂Ω

ψ ∈ Rγ,β ,

there exists a weak solution [u, z, w] of problem (Sγ,βφ,ψ).

In the special case R−γ,β = R+γ,β , that is, when γ(r) = β(r) = 0 for any

r ∈ R, existence and uniqueness of weak solutions are also obtained.

Theorem 3.4 (([2]). For any φ ∈ V 1,p(Ω) and ψ ∈ V 1,p(∂Ω) with∫Ω

φ+∫∂Ω

ψ = 0,

there exists a unique (up to a constant) weak solution u ∈ W 1,p(Ω) of theproblem

−div a(x,Du) = φ in Ω

a(x,Du) · η = ψ on ∂Ω

in the sense that ∫Ω

a(x,Du) ·Dv =∫∂Ω

ψv +∫

Ω

φv,

for all v ∈W 1,p(Ω).

In the case ψ = 0 we have the following result without imposing any condi-tion on γ, in the same line to the one obtained by Bénilan, Crandall and Sackin [17] for the Laplacian operator and L1-data.

Theorem 3.5 (([2]). Assume D(β) = R or a smooth, and R−γ,β < R+γ,β .

(i) For any φ ∈ V 1,p(Ω) such that∫

Ωφ ∈ Rγ,β , there exists a weak solution

[u, z, w] of problem (Sγ,βφ,0 ), with z << φ.

(ii) For any [u1, z1, w1] weak solution of problem (Sγ,βφ1,0), φ1 ∈ V 1,p(Ω),∫

Ωφ1 ∈ Rγ,β , and any [u2, z2, w2] weak solution of problem (Sγ,βφ2,0

),φ2 ∈ V 1,p(Ω),

∫Ωφ2 ∈ Rγ,β , we have that∫

Ω

(z1 − z2)+ +∫∂Ω

(w1 − w2)+ ≤∫

Ω

(φ1 − φ2)+.


For Dirichlet boundary condition we have the following result.

Theorem 3.6 (([2]). Assume D(β) = 0. For any φ ∈ V 1,p(Ω), there exists aunique [u, z] = [uφ,ψ, zφ,ψ] ∈W 1,p

0 (Ω)× V 1,p(Ω), z ∈ γ(u) a.e. in Ω, such that∫Ω

a(x,Du) ·Dv +∫

Ω

zv =∫

Ω

φv,

for all v ∈W 1,p0 (Ω).

Moreover, if φ1, φ2 ∈ V 1,p(Ω), then∫Ω

(zφ1,ψ1 − zφ2,ψ2)+ ≤∫

Ω

(φ1 − φ2)+.

Let us now state the existence results of entropy solutions for data in L1.

Theorem 3.7 (([2, 5]). Assume D(γ) = R and either D(β) = R or a smooth.Then,

(i) for any φ ∈ L1(Ω) and ψ ∈ L1(∂Ω), there exists an entropy solution[u, z, w] of problem (Sγ,βφ,ψ).

(ii) For any [u1, z1, w1] entropy solution of problem (Sγ,βφ1,ψ1), φ1 ∈ L1(Ω),

ψ1 ∈ L1(∂Ω), and any [u2, z2, w2] entropy solution of problem (Sγ,βφ2,ψ2),

φ2 ∈ L1(Ω), ψ2 ∈ L1(∂Ω), we have that∫Ω

(z1 − z2)+ +∫∂Ω

(w1 − w2)+ ≤∫∂Ω

(ψ1 − ψ2)+ +∫

Ω

(φ1 − φ2)+.

As a consequence of the previous results we have the following corollary(see [17, Propostion C (iv)] for the Laplacian).

Corollary 3.1. a is smooth if and only if for any φ ∈ L1(Ω) there existsT (φ) ∈ L1(∂Ω) such that the entropy solution u of

−div a(x,Du) = φ in Ω

u = 0 on ∂Ω,


is an entropy solution of−div a(x,Du) = φ in Ω

a(x,Du) · η = T (φ) on ∂Ω.

Moreover, the map T : L1(Ω)→ L1(∂Ω) satisfies∫Ω

(T (φ1)− T (φ2))+ ≤∫

Ω

(φ1 − φ2)+,

for all φ1, φ2 ∈ L1(Ω), and T(V 1,p(Ω)

)⊂ V 1,p(∂Ω).

In the homogeneous case, without any condition on γ, we also get thefollowing result.

Theorem 3.8 (([2, 5]). Assume D(β) = R or a is smooth. Then,

(i) for any φ ∈ L1(Ω), there exists an entropy solution [u, z, w] of problem(Sγ,βφ,0 ), with z << φ.

(ii) For any [u1, z1, w1] entropy solution of problem (Sγ,βφ1,0), φ1 ∈ L1(Ω), and

any [u2, z2, w2] entropy solution of problem (Sγ,βφ2,0), φ2 ∈ L1(Ω), we have

that ∫Ω

(z1 − z2)+ +∫∂Ω

(w1 − w2)+ ≤∫

Ω

(φ1 − φ2)+.

In order to obtain the existence results the main idea is to consider theapproximate problems

(Sγm,n,βm,nφm,n,ψm,n)

−div a(x,Du) + γm,n(u) 3 φm,n in Ω

a(x,Du) · η + βm,n(u) 3 ψm,n on ∂Ω,

where γm,n and βm,n are approximations of γ and β given by

γm,n(r) = γ(r) +1mr+ − 1

nr−

andβm,n(r) = β(r) +

1mr+ − 1

nr−


respectively, m,n ∈ N, and

φm,n = supinfm,φ,−n

andψm,n = supinfm,ψ,−n

are approximations of φ and ψ respectively. For these approximate problems weobtain existence of weak solutions with appropriated estimates and monotoneproperties, which allow us to pass to the limit.

4. An obstacle problem

Let us now suppose that

R 6= D(γ) ⊂ D(β).

Observe that if D(γ) is not bounded we are dealing with a one obstacle problemand with a two obstacle problem if D(γ) is bounded.

We want to stress, as remarked in the introduction, that for this kind ofproblems the existence of weak solution, in the usual sense, fails to be truefor nonhomogeneous boundary conditions. We introduce a new notion of solu-tion for which existence and uniqueness can be obtained for nonhomogeneousboundary conditions in the case D(γ) 6= R.

To give an idea of this new concept of solution, let us consider again theproblem (Lγ,0φ,ψ) treated in the introduction, with φ ∈ Lp′

(Ω), ψ ∈ Lp′(∂Ω) and

D(γ) = [0, 1]. In order to get existence, it is usual to use the approximateproblems

(Lγr,0φ,ψ )

−∆ur + γr(ur) = φ in Ω

∂ηur = ψ on ∂Ω,

where γr is the Yosida approximation of γ. Now, by the results in the previoussection, the estimates we can obtain are, essentially, γr(ur) bounded in L1(Ω)and ur bounded in H1(Ω). Therefore we have to pass to the limit weakly-star in the space of measures for γr(ur) and weakly in H1(Ω) for ur. And the


standard analysis for this kind of problems allows us to obtain a couple [u, µ],where u ∈ H1(Ω), µ is a diffuse Radon measure in RN concentrated in Ω and∫

Ω

Du ·Dv +∫

Ω

v dµ+∫∂Ω

v dµ =∫

Ω

φv +∫∂Ω

ψv ∀ v ∈ H1(Ω) ∩ L∞(Ω).

In a first step, we prove that the Radon-Nykodym decomposition of µ relativelyto the Lebesgue measure, µ = µa + µs, is such that µa ∈ γ(u) a.e. in Ω and µsis concentrated on x ∈ Ω ; u(x) = 0 ∪ x ∈ Ω ; u(x) = 0 with

µs ≤ 0 on x ∈ Ω ; u(x) = 0 and µs ≥ 0 on x ∈ Ω ; u(x) = 1.

Afterwards, an accurate analysis allows us to prove moreover that µs is con-centrated on ∂Ω and it is absolutely continuous with respect to an integrablefunction on the boundary, which implies that µs ∈ L1(∂Ω). So, [u, µa, µs] is aweak solution, in the usual sense, of the problem

−∆u+ γ(u) 3 φ in Ω

Du · η + ∂I[0,1](u) 3 ψ on ∂Ω,

where, for an interval I ⊂ R, ∂II denotes the subdifferential of the indicatorfunction of I,

II(r) =

0 if r ∈ I,

+∞ if r /∈ I,which is the maximal monotone graph defined by D(∂II) = I and ∂II(r) =0 for r ∈ int(I). For instance, if D(γ) = [a, b], then

∂ID(γ)(r) =

]−∞, 0] if r = a

0 if a < r < b

[0,+∞[ if r = b.

In other words, the boundary condition needs to be fullfield in the followingsense

(4.1)

∂ηu = ψ on [0 < u < 1]∂ηu ≥ ψ on [u = 1]∂ηu ≤ ψ on [u = 0].


The last two conditions of (4.1) disappear whenever the data φ and ψ are suchthat the sets [u = 1] and [u = 0] are negligeable. This is the case, for instance,if ψ ≡ 0.

After a complete analysis of the general problem (Sγ,βφ,ψ) when D(γ) ⊂ D(β),we get the right notion of solution which coincides with the concept of weaksolution for the problem

−div a(x,Du) + γ(u) 3 φ in Ω

a(x,Du) · η + β(u) + ∂ID(γ)

(u) 3 ψ on ∂Ω.

Definition 4.1. Let φ ∈ L1(Ω) and ψ ∈ L1(∂Ω). A triple of functions[u, z, w] ∈W 1,p(Ω)×L1(Ω)×L1(∂Ω) is a generalized weak solution of problem(Sγ,βφ,ψ) if z(x) ∈ γ(u(x)) a.e. in Ω, w(x) ∈ β(u(x)) + ∂I

D(γ)(u(x)) a.e. on ∂Ω,

and ∫Ω

a(x,Du) ·Dv +∫

Ω

zv +∫∂Ω

w v =∫

Ω

φv +∫∂Ω

ψv,


It is clear that a weak solution is a generalized weak solution. Moreover,thanks to the results in Section 3, the two concepts coincide in the case (A).Let us show the existence and uniqueness results of solutions in the sense ofDefinition 4.1 for (Sγ,βφ,ψ) in the case R 6= D(γ) ⊂ D(β).

The main result about existence is divided in two statements. The state-ment (i) corresponds to the existence of generalized weak solutions for the oneor two obstacle problem and regular data. Observe that for the one obsta-cle problem Rγ,β can be different from R and for the two obstacle problemRγ,β = R. The statement (ii) is for the two obstacle problem and L1-data.

Theorem 4.1 (([4]). Assume R 6= D(γ) ⊂ D(β). Then,

(i) for any φ ∈ V 1,p(Ω) and ψ ∈ V 1,p(∂Ω) such that∫Ω

φ+∫∂Ω

ψ ∈ Rγ,β ,

there exists a generalized weak solution [u, z, w] of problem (Sγ,βφ,ψ);


(ii) if D(γ) is bounded, the existence of a generalized weak solution [u, z, w]of problem (Sγ,βφ,ψ) holds to be true for any φ ∈ L1(Ω) and ψ ∈ L1(∂Ω).

Since [u, z, w] is a generalized weak solution of problem (Sγ,βφ,ψ) if and onlyif [u, z, w] is a weak solution of problem (Sγ,βγφ,ψ ), where βγ := β + ∂I

D(γ), the

uniqueness of a generalized weak solution is a consequence of Theorems 3.1 and3.2.

Theorem 4.2. Let φ ∈ L1(Ω) and ψ ∈ L1(∂Ω). Let [u1, z1, w1] and [u2, z2, w2]be generalized weak solutions of (Sγ,βφ,ψ). Then, there exists a constant c ∈ Rsuch that

u1 − u2 = c a.e. in Ω,

z1 − z2 = 0 a.e. in Ω

andw1 − w2 = 0 a.e. in ∂Ω.

If c 6= 0, z1 = z2 is constant.Moreover, given [u1, z1, w1] a generalized weak solution of (Sγ,βφ1,ψ1

) and[u2, z2, w2] a generalized weak solution of (Sγ,βφ2,ψ2

),∫Ω

(z1 − z2)+ +∫∂Ω

(w1 − w2)+ ≤∫

Ω

(φ1 − φ2)+ +∫∂Ω

(ψ1 − ψ2)+.

By the above result we have that [0, φ, ψ] is the unique generalized weaksolution of the paradigmatic problem (Lγ,0φ,ψ).

Thanks to the example (Lγ,0φ,ψ) it is clear that, for a generalized weak solu-tion, w /∈ β(u) in general. However we show that for the homogeneous caseψ ≡ 0, w ∈ β(u).

In order to prove the existence results we define

Mpb(Ω) :=

µ ∈Mb(UΩ) ∩W−1,p′

(UΩ) : µ is concentrated on Ω.

This definition is independent of the open set UΩ, and for u ∈ W 1,p(Ω) andµ ∈Mp

b(Ω) we have

〈µ, u〉 =∫

Ω

u dµ+∫∂Ω

u dµ.


Observe also that if µ ∈Mpb(Ω) then µ ∈Mp

b(UΩ).We also define, for u ∈W 1,p(Ω) and µ ∈Mp

b(Ω),

µ ∈ γ(u)

if the following conditions are satisfied,

(i) µa ∈ γ(u) LN − a.e. in Ω,

(ii) µs is concentrated on the set x ∈ Ω : u = γ(i) ∪ x ∈ Ω : u = γ(s),

(iii) µs ≤ 0 on x ∈ Ω : u = γ(i) and µs ≥ 0 on x ∈ Ω : u = γ(s),

where γ(i) = inf Dom(γ) and γ(s) = supDom(γ).Let us point out the similitude of the above concept with the definition of

µ ∈ γ(u) for µ ∈Mpb(Ω) given in [9], from where we have taken some ideas for

our proofs.Let u ∈W 1,p(Ω) and µ ∈Mp

b(Ω). Recalling that the set x ∈ Ω : u = ±∞has zero p-capacity relative to UΩ and that µ ∈Mp

b(UΩ), we have that

µs = 0 on x ∈ Ω : u = ±∞.

In particular, if D(γ) = R, then

µ ∈ γ(u) if and only if µ ∈ L1(Ω) and µ ∈ γ(u) a.e. in Ω.

Initially we prove the following proposition by approximating the problemby other one to which we can apply the results in Section 3. Concretely,we consider approximate problems replacing β by β, being β the maximalmonotone graph defined by

β(s) =

β(s) if s ∈]γ(i), γ(s)[,

β0(γ(i)) if s < γ(i) (γ(i) finite),

β0(γ(s)) if s > γ(s) (γ(s) finite).

And replacing γ by γr as follows, in the case domain of γ is bounded, i.e. γ(i)

and γ(s) are both finite, for every r ∈ N we take γr = γr to be the Yosidaapproximation of γ, and in the case D(γ) is not bounded, we consider that


γ(i) = −∞ and γ(s) is finite (the other case, γ(i) finite and γ(s) = +∞, beingsimilar), for every r ∈ N we take γr the maximal monotone graph defined by

γr(s) =

γ(s) if s < 0,

γr(s) if s > 0.

It is clear that in the last case we are regularizing just the positive part, theregularization of the negative part is not necessary since it is everywhere de-fined.

Proposition 4.1 (([4]). Assume R 6= D(γ) ⊂ D(β). Then, for any φ ∈ V 1,p(Ω)and ψ ∈ V 1,p(∂Ω) such that ∫

Ω

φ+∫∂Ω

ψ ∈ Rγ,β ,

there exists [u, µ,w] ∈ W 1,p(Ω) ×Mpb (Ω) × V 1,p(∂Ω) such that w ∈ β(u) a.e.

in ∂Ω, µ ∈ γ(u) and∫Ω

a(x,Du) ·Dv + 〈µ, v〉+∫∂Ω

wv =∫

Ω

φv +∫∂Ω

ψv ∀ v ∈W 1,p(Ω).

Next we show that µs is concentrated on ∂Ω and is absolutely continuousrespect to an integrable function in order to get Theorem 4.1. To this end, thefollowing technical result is established.

Lemma 4.1. Let η ∈ W 1,p(Ω), ν ∈ Mpb (Ω) and λ ∈ R be such that η ≤ λ

(resp. η ≥ λ) a.e. Ω. If−div a(x,Dη) = ν

in the sense that∫

Ω

a(x,Dη) ·Dξ =∫

Ω

ξ dν, for any ξ ∈W 1,p(Ω), then

∫x∈Ω:η(x)=λ

ξ dν ≥ 0

(resp. ∫x∈Ω:η(x)=λ

ξ dν ≤ 0)

for any ξ ∈W 1,p(Ω), ξ ≥ 0.


5. Measure data

We define

Mpb(Ω) :=

µ ∈Mp

b(UΩ) : µ is concentrated on Ω.

This definition is independent of the open set UΩ. Note that for u ∈W 1,p(Ω)∩L∞(Ω) and µ ∈Mp

b(Ω), we have

〈µ,E(u)〉 =∫

Ω

u dµ+∫∂Ω

u dµ;

on the other hand, there exists f ∈ L1(UΩ) and F ∈ (Lp′(UΩ))N such that

µ = f + div(F ), therefore, we also can write

〈µ,E(u)〉 =∫UΩ

fE(u) dx−∫UΩ

F ·DE(u) dx.

Note that, if f ∈ L1(Ω) and g ∈ L1(∂Ω) then fLN Ω + gHN−1 ∂Ω is adiffuse measure concentrated in Ω. Now, if p > N−k, 1 ≤ k < N−1, and M isa k-rectifiable subset of ∂Ω, then Hk M is a diffuse measure concentrated in∂Ω which is not an L1 function in ∂Ω (see, [39, Theorem 2.26] or [47, Theorem2.6.16]).

Definition 5.1. Let µ1, µ2 measures, µ1 = µ1 Ω and µ2 = µ2 ∂Ω, such thatµ1 + µ2 ∈Mp

b(Ω). A triple of functions [u, z, w] ∈ W 1,p(Ω)× L1(Ω)× L1(∂Ω)is a weak solution of problem (Sγ,βµ1,µ2

) if z(x) ∈ γ(u(x)) a.e. in Ω, w(x) ∈β(u(x)) a.e. in ∂Ω and∫

Ω

a(x,Du) ·Dv dx+∫

Ω

zv dx+∫∂Ω

wv dHN−1 =∫

Ω

v dµ1 +∫∂Ω

v dµ2

for all v ∈W 1,p(Ω) ∩ L∞(Ω).

As we pointed out before the concept of weak solution for this type ofproblems is not enough in order to get uniqueness. It is necessary to findsome extra conditions on the distributional solutions in order to ensure bothexistence and uniqueness. This was done by introducing the concepts of entropyand renormalized solutions (see, e.g., [12]). For our problem these concepts arethe following.


Definition 5.2. Let µ1, µ2 measures, µ1 = µ1 Ω and µ2 = µ2 ∂Ω, such thatµ1 + µ2 ∈ Mp

b(Ω). A triple of functions [u, z, w] ∈ T 1,ptr (Ω) × L1(Ω) × L1(∂Ω)

is an entropy solution of problem (Sγ,βµ1,µ2) if z(x) ∈ γ(u(x)) a.e. in Ω, w(x) ∈

β(u(x)) a.e. in ∂Ω and∫Ω

a(x,Du) ·DTk(u− v) dx+∫

Ω

zTk(u− v) dx

+∫∂Ω

wTk(u− v) dHN−1

≤∫

Ω

Tk(u− v) dµ1 +∫∂Ω

Tk(u− v) dµ2 ∀k > 0,

(5.1)

for all v ∈W 1,p(Ω) ∩ L∞(Ω).

Definition 5.3. Let µ1, µ2 measures, µ1 = µ1 Ω and µ2 = µ2 ∂Ω, such thatµ1 + µ2 ∈ Mp

b(Ω). A triple of functions [u, z, w] ∈ T 1,ptr (Ω) × L1(Ω) × L1(∂Ω)

is a renormalized solution of problem (Sγ,βµ1,µ2) if z(x) ∈ γ(u(x)) a.e. in Ω,

w(x) ∈ β(u(x)) a.e. in ∂Ω, and the following conditions hold

(a) for every h ∈W 1,∞(R) with compact support we have∫Ω

a(x,Du) ·Duh′(u)ϕdx+∫

Ω

a(x,Du) ·Dϕh(u) dx

+∫

Ω

z h(u)ϕdx+∫∂Ω

w h(u)ϕdHN−1

=∫

Ω

h(u)ϕdµ1 +∫∂Ω

h(u)ϕdµ2 ∀k > 0,

(5.2)

for all ϕ ∈W 1,p(Ω) ∩ L∞(Ω) such that h(u)ϕ ∈W 1,p(Ω),

(b)

limn→+∞

∫n≤|u|≤n+1

a(x,Du) ·Dudx = 0.

Remark 5.1 - Every term in (5.2) is well defined. This is clear for the righthand side since h(u)ϕ belongs to L∞(Ω, µ1 + µ2), and thus to L1(Ω, µ1 + µ2).


On the other hand, since supp(h) ⊂ [−k, k] for some k > 0, the two first termsof the left hand side can be written as∫

Ω

a(x,DTk(u)) ·DTk(u)h′(u)ϕdx+∫

Ω

a(x,DTk(u)) ·Dϕh(u) dx,

and both integrals are well defined in view of (H2), since both ϕ and Tk(u)belong to W 1,p(Ω). Moreover, it is not difficult to see that the productDTk(u)h′(u) coincides with the gradient of the composite function h(u) =h(Tk(u)) almost everywhere (see [21]).

The nexus relating both concepts of solutions is the following one.

Lemma 5.1. Let µ1, µ2 measures, µ1 = µ1 Ω and µ2 = µ2 ∂Ω, such thatµ1 + µ2 ∈ Mp

b(Ω). Let [u, z, w] be an entropy solution of problem (Sγ,βµ1,µ2).

Then,

limh→+∞

∫x∈Ω:h<|u(x)|<h+k

|Du|p = 0, ∀ k > 0.

Theorem 5.1. Let µ1, µ2 measures, µ1 = µ1 Ω and µ2 = µ2 ∂Ω, such thatµ1 + µ2 ∈Mp

b(Ω). Then, [u, z, w] is an entropy solution of problem (Sγ,βµ1,µ2) if

and only if [u, z, w] is a renormalized solution of problem (Sγ,βµ1,µ2).

Remark 5.2 - Assume that [u, z, w] is an entropy solution of problem (Sγ,βµ1,µ2).

If we take v = Th(u)± 1 as test functions in (5.1) and let h go to +∞, we getthat ∫

Ω

z +∫∂Ω

w = µ1(Ω) + µ2(∂Ω).

Then, since z(x) ∈ γ(u(x)) a.e. in Ω and w(x) ∈ β(u(x)) a.e. in ∂Ω, necessarilyµ1 and µ2 must satisfy

R−γ,β ≤ µ1(Ω) + µ2(∂Ω) ≤ R+γ,β .

For weak solutions a contraction principle is proved in Theorem 5.3. Respectto uniqueness for entropy solutions we have the following general result.

Theorem 5.2 (([5]). Let µ1, µ2 measures, µ1 = µ1 Ω and µ2 = µ2 ∂Ω, suchthat µ1 + µ2 ∈Mp

b(Ω). Let [u1, z1, w1] and [u2, z2, w2] be entropy solutions ofproblem (Sγ,βµ1,µ2

). Then, there exists a constant c ∈ R such that

u1 − u2 = c a.e. in Ω,


z1 − z2 = 0 a.e. in Ω.

w1 − w2 = 0 a.e. in ∂Ω.

Moreover, if c 6= 0, there exists a constant k ∈ R such that z1 = z2 = k.

In order to get the existence of solutions we use the following key results.

Lemma 5.2 (([5]). Let Ω be a bounded domain in RN with ∂Ω of class C1.Given µ ∈Mp

b(Ω), there exists a sequence ψnn∈N ⊂ Cc(Ω),

ψn µ as measures,

such that, for any vnn∈N ∈W 1,p(Ω) with vn → v weakly in W 1,p(Ω) and allk > 0,

limn→∞

∫Ω

Tk(vn)ψn =∫

Ω

Tk(v) dµ.

Moreover, if µ = f + divF , f ∈ Lp′(UΩ), F ∈ Lp′

(UΩ)N , then∣∣∣∣∫Ω

vnψn

∣∣∣∣ ≤ C1 ‖f‖Lp′ (UΩ) ‖vn‖Lp(Ω)

+ C2‖F‖(Lp′ (UΩ))N

(‖vn‖Lp(Ω) + ‖Dvn‖Lp(Ω)

).

Grosso modo, we rectify ∂Ω using local maps (Gi, Ui)i=0,1,...,k and in-troduce a partition of unit θii=0,1,...,k subordinate to ∂Ω and U1, . . . Uk (see[24]). Letting dε(x′, xN ) = (x′, (1− ε)xN + ε) and writing din = Gi d 1

nGi−1,

i = 1, ..., n, we take

(5.3) ψn =

(k∑i=1

din#σi + σ0

)∗ ρ 1

2n,

where ρε is a mollifier with support in B(0, ε) and din#ν is the push-forwardmeasure.

Lemma 5.3 (([1]). Let unn∈N ⊂ W 1,p(Ω), znn∈N ⊂ L1(Ω), wnn∈N ⊂L1(∂Ω) such that, for every n ∈ N, zn ∈ γ(un) a.e. in Ω and wn ∈ β(un) a.e.in ∂Ω. Let us suppose that


(i) if R+γ,β = +∞, there exists M > 0 such that∫

Ω

z+n dx+

∫∂Ω

w+n dσ < M ∀n ∈ N;

(ii) if R+γ,β < +∞, there exists M ∈ R such that∫

Ω

zndx+∫∂Ω

wndσ < M < R+γ,β

and

limL→+∞

(∫x∈Ω:zn(x)<−L

|zn|dx+∫x∈∂Ω:wn(x)<−L

|wn|dσ

)= 0

uniformly in n ∈ N. Then, there exists a constant C = C(M) such that

‖u+n ‖Lp(Ω) ≤ C

(‖Du+

n ‖Lp(Ω) + 1)

∀n ∈ N.

The next theorem gives the existence results of weak and entropy solutions.A contraction principle for weak sub and super solutions also holds.

Theorem 5.3 (([5]). Let Ω ⊂ RN be an open bounded set with boundary ∂Ωof class C1. Assume Dom(γ) = Dom(β) = R and J(γ) and J(β) are bounded.Then,

(i) for any measures µ1, µ2 such that µ1 = µ1 Ω, µ2 = µ2 ∂Ω, µ = µ1 +µ2 = f + div(F ), f ∈ Lp′

(UΩ), F ∈ (Lp′(UΩ))N , and µ(Ω) ∈ Rγ,β , there

exists a weak solution [u, z, w] of problem (Sγ,βµ1,µ2).

(ii) If [u, z, w] is a weak subsolution of problem (Sγ,βµ1,µ2), µ1, µ2 measures,

µ1 = µ1 Ω and µ2 = µ2 ∂Ω such that µ = µ1 + µ2 ∈ Mpb(Ω), and

[u, z, w] is a weak supersolution of problem (Sγ,βµ1,µ2), µ1, µ2 measures,

µ1 = µ1 Ω and µ2 = µ2 ∂Ω such that µ = µ1 + µ2 ∈Mpb(Ω), then

(5.4)∫

Ω

(z − z)+ +∫∂Ω

(w − w)+ ≤ (µ− µ)+(Ω).

(iii) For any measures µ1, µ2 such that µ1 = µ1 Ω, µ2 = µ2 ∂Ω, µ =µ1 + µ2 ∈ Mp

b(Ω), and µ(Ω) ∈ Rγ,β , there exists an entropy solution[u, z, w] of problem (Sγ,βµ1,µ2

).


For the existence results we first suppose that Rγ,β = R and consider asequence of approximated problems, replacing ψ by ψn, given in (5.3), to whichwe can apply Theorem 3.5. Afterwards we use monotonicity arguments to dealwith the other cases.

The proof of the contraction principle is as follows. Let [u, z, w] be a weaksubsolution of problem (Sγ,βµ1,µ2

) and let [u, z, w] be a weak supersolution ofproblem (Sγ,βµ1,µ2

). Then,

∫Ω

a(x,Du) ·Dv +∫

Ω

zv +∫∂Ω

wv ≤∫

Ω

v dµ

∀ v ∈W 1,p(Ω) ∩ L∞(Ω), v ≥ 0,

(5.5)

and ∫Ω

a(x,Du) ·Dv +∫

Ω

zv +∫∂Ω

wv ≥∫

Ω

v dµ

∀ v ∈W 1,p(Ω) ∩ L∞(Ω), v ≥ 0.

(5.6)

Consider ρ ∈W 1,p(Ω), 0 ≤ ρ ≤ 1. Taking as test function v = T+k

k (u− u+ kρ)in (5.5) and in (5.6), we have

∫Ω

a(x,Du) ·D(T+k

k(u− u+ kρ)

)+∫

Ω

zT+k

k(u− u+ kρ)

+∫∂Ω

wT+k

k(u− u+ kρ) ≤

∫Ω

(T+k

k(u− u+ kρ)

)dµ

and ∫Ω

a(x,Du) ·D(T+k

k(u− u+ kρ)

)+∫

Ω

zT+k

k(u− u+ kρ)

+∫∂Ω

wT+k

k(u− u+ kρ) ≥

∫Ω

(T+k

k(u− u+ kρ)

)dµ


Therefore, having in mind the monotonicity of a, we get∫Ω

(z − z)T+k

k(u− u+ kρ) +

∫∂Ω

(w − w)T+k

k(u− u+ kρ)

+∫0<u−u+kρ<k

(a(x,Du)− a(x,Du)) ·Dρ

≤∫

Ω

(T+k

k(u− u+ kρ)

)d(µ− µ) ≤ (µ− µ)+(Ω).

Taking limit when k goes to 0 in the above expression, having in mind thatDu1 = Du2 where u1 = u2, we obtain that

(5.7)

∫Ω

(z − z)sign+0 (u− u)χu6=u +

∫Ω

(z − z)ρχu=u

∫∂Ω

(w − w)sign+0 (u− u)χu6=u +

∫∂Ω

(w − w)ρχu=u

≤ (µ− µ)+(Ω).

By approximation we can suppose that (5.7) holds for every 0 ≤ ρ ∈W 1,1(Ω)∩L∞(Ω). It is easy to see that there exist 0 ≤ ρn ∈W 1,1(Ω)∩L∞(Ω) such thatρn = sign+

0 (w − w) HN−1 a.e. on ∂Ω and

ρn → sign+0 (z − z) in L1(Ω).

Then, taking ρ = ρn in (5.7) and sending n → +∞, we get the contractionprinciple (5.4).

Remark 5.3 - We also get the following monotonicity result for entropy solu-tions.Let [u, z, w] be an entropy solution of problem (Sγ,βµ1,µ2

), µ1, µ2 measures, µ1 =µ1 Ω and µ2 = µ2 ∂Ω such that µ = µ1 +µ2 ∈Mp

b(Ω) and µ(Ω) ∈ Rγ,β , and[u, z, w] an entropy solution of problem (Sγ,βµ1,µ2

), µ1, µ2 measures, µ1 = µ1 Ωand µ2 = µ2 ∂Ω such that µ = µ1+µ2 ∈Mp

b(Ω) and µ(Ω) ∈ Rγ,β . If µ1 ≤ µ2,then z1 ≤ z2 a.e. in Ω and w1 ≤ w2 a.e. in ∂Ω.


Remark 5.4 - We point out that the cases of Hele Shaw problem, which corre-sponds to

γ(r) =

0 if r < 0,[0, 1] if r = 0,1 if r > 0,

and the multiphase Stefan problem, which corresponds to

γ(r) =

r − 1 if r < 0,[−1, 0] if r = 0,r if r > 0,

are included in the above existence and uniqueness results.

In the particular case of Dirichlet boundary condition, that is, for β themonotone graph D = 0 × R, which corresponds to the problem

(Sγ,Dµ,0 )

−div a(x,Du) + γ(u) 3 µ in Ω

u = 0 on ∂Ω,

where γ is a maximal monotone graph in R2 and µ a diffuse measure in Ω, wealso have existence and uniqueness of entropy solutions.

6. Applications

One of the main applications of these results is the study of doubly non-linear evolution problems of elliptic-parabolic type and degenerate parabolicproblems of Stefan or Hele-Shaw type, with nonhomogeneous boundary condi-tions and/or dynamical boundary conditions.

The results obtained have an interpretation in terms of accretive operators.Indeed, we can define the (possibly multivalued) operator Bγ,β in X := L1(Ω)×L1(∂Ω) as

Bγ,β :=

((v, w), (v, w)) ∈ X ×X : ∃u ∈ T 1,ptr (Ω),

with [u, v, w] an entropy solution of (Sγ,βv+v,w+w).


Then, under certain assumptions, Bγ,β is an m-T-accretive operator in X.Therefore, by the theory of Evolution Equations Governed by Accretive Op-erators (see, [11], [16] or [29]), for any (v0, w0) ∈ D(Bγ,β)

Xand any (f, g) ∈

L1(0, T ;L1(Ω)) × L1(0, T ;L1(∂Ω)), there exists a unique mild-solution of theproblem

V ′ + Bγ,β(V ) 3 (f, g), V (0) = (v0, w0),

which rewrites, as an abstract Cauchy problem in X, the following degenerateelliptic-parabolic problem with nonlinear dynamical boundary conditions

(DP )

vt − diva(x,Du) = f, v ∈ γ(u), in Ω× (0, T )

wt + a(x,Du) · η = g, w ∈ β(u), on ∂Ω× (0, T )

v(0) = v0 in Ω, w(0) = w0 in ∂Ω.

In principle, it is not clear how these mild solutions have to be interpretedrespect to the problem (DP ). In [1] and [3] we characterize these mild solutionsas weak solutions or as entropy solutions depending on the regularity of thedata.

Acknowledgements. The first, third and fourth authors have been partiallysupported by the Spanish MEC and FEDER, project MTM2005-00620.

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