Perron's method for quasilinear hyperbolic systems: Part I

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milin-ations,xistence

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J. Math. Anal. Appl. 316 (2006) 357–368

www.elsevier.com/locate/jma

Perron’s method for quasilinear hyperbolic systemPart I

Penelope Smith

Lehigh University, Bethlehem, PA 18015, USA

Received 21 April 2005

Available online 19 May 2005

Submitted by William F. Ames

Abstract

We define a notion of viscosity solution (sub-, supersolution) for these systems, prove a coson principle and we prove existence of viscosity solutions using a Perron like method. In Pado all the above except prove existence using the Perron method. 2005 Elsevier Inc. All rights reserved.

Keywords:Hyperbolic systems; Viscosity solutions

1. Introduction

In several previous papers [1–3] the author proved a comparison principle for a seear second order wave equation, defined a notion of viscosity solution for such equand showed that Perron’s method extended to these hyperbolic equations to prove eof viscosity solutions of the Cauchy problem for these equations.

Such equations can be written as first order symmetric hyperbolic systems [4,5].paper, we prove comparison theorems for sub- and supersolutions of first order symhyperbolic systems, define a notion of viscosity solution for these systems, and exteron’s method of upper and lower envelopes to prove the existence of continuous vissolutions for the long time and eternal Cauchy problem for such systems.

E-mail address:[email protected].

0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2005.04.055

358 P. Smith / J. Math. Anal. Appl. 316 (2006) 357–368

d tohysics[5–7]

pendentnce oflem.in thisinuous

linear,d

ctly

the

. Let

m

Our motivation is provided by the fact that this will yield a new numerical methosolve such systems which include most of the important equations of mathematical pand, in particular, the long term Cauchy problem for Einstein field equations, whichcan be shown to be equivalent to solving such a system.

Indeed, in a second paper, we use the results of this paper, and a gauge indemethod, due to Deturk [7], of showing the above equivalence, to show the existelong term, possible only continuous, viscosity solutions of the Einstein–Cauchy prob

We also note that the Navier–Stokes equations (for supersonic flow) can be putform, and we have results in progress on the general long term existence for contviscosity solutions of these equations that use the results of this paper.

2. Quasilinear first order symmetric hyperbolic systems

For the convenience of the reader, we recall the definition of a first order, quasisymmetric hyperbolic system in a slab domainDT of R+ ×Rn. More details may be founin [5,8].

Definition 1. Let n,N ∈ Z+. Let DT := [0, T ] × Rn.Let u :R+ × Rn → RN .Let A0 :R+ × Rn × RN → Sym+(N,N) beC∞ (in fixed coordinates,A0 is anN × N

symmetric matrix), and letA0 be positive definite, that is: all of its eigenvalues are stribigger than zero.

Let i = 1,2, . . . , n. For eachi, let

Ai :R+ × Rn × RN → Sym(N,N)

be C∞ and inC2+ε0 ∃ε0 ∈ (0,1), where we require a uniform Hölder constant forwhole domainDT (in a fixed coordinates eachAi is a symmetric matrix). Let SupDT

|Ai |,|Dx,tA

i |, |D2x,tA

i | < ∞. Let ∂iu and∂0u exist inD0T .

Definition 2. Let B :R+ × Rn → L(N,N) beC∞. Let all the off diagonal terms ofB benonnegative. We call suchAi admissible.

Definition 3. Let f :R+ ×Rn → RN beC∞, letf (t, x) 0, and let lim|x|→∞ f (t, x) = 0.Let f ∈ H 1,2(Rn,RN), where we mean that one weak derivative is square integrablef ∈ Cε0(DT ,RN), with ε0 as above. We call suchf admissible.

Definition 4. We say thatu satisfies afirst order quasilinear symmetric hyperbolic systein D0

T iff: in D0T :

Lu := A0(t, x,u)∂tu + Ai(t, x,u)∂iu − B(t, x)u − f (t, x) = 0.

(Here repeated indices are summed.) Also we requireAi andf to be admissible.

Note that a.n.l.o.g., by standard methods [5,8], we can assume thatA0 = I . We do thisfrom now on, for simplicity of presentation.

P. Smith / J. Math. Anal. Appl. 316 (2006) 357–368 359

tion ofctions.

.ooth-ooth

condingularut ap-

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ctionsr

ains.

3. Smoothing and approximations of vector functions

In order to define our notions of generalized solution, subsolution and supersoluthe above systems, we need to use smoothing of vector valued semicontinuous funOur smoothings will always be done componentwise for our vector valued functions

We need to combine two smoothings, one given by convolution with a Poisson sming kernel onR1×Rn, and the other by a more sophisticated smoothing used in nonsmanalysis and in the theory of elliptic and parabolic viscosity solutions [9,10]. This setype of smoothing, based on quadratic envelopes, has good limit properties at spoints of semicontinuous functions. It approaches the singular values pointwise, bplied to a semicontinuous function it only gives a Lipschitz function.

Thus, we will first smooth our semicontinuous components by this quadratic envsmoothing (denoted by a positive parameterγ ), and then we will smooth the result byPoisson smoothing (denoted by a positive parameterσ ).

The result of this double smoothing will be aC∞ vector function depending on thtwo positive parametersγ ,σ , with the property that, when we first letσ ↓ 0, and then leγ ↓ 0, this function will converge pointwise in each component to our given semiconous function.

This nice behavior at discontinuous points will allow us to simplify and generalize sof the proofs in the author’s previous papers [1–3].

3.1. Extension of vector functions

In order to construct our smoothings we first need to extend semicontinuous fun(each component separately) from a regular domainDT to all of R ×RN , in such a mannethat the extension is compactly supported in the time variable (i.e. inR).

We assume throughout that our vector functionu is in L1(DT ,RN) ∩ L∞(DT ,RN).Denote its components byui , wherei = 1,2, . . . ,N .Let ui :DT → R. We first extendui to the doubleDT,Double obtained by reflection.For all X ∈ DT let X =: (t, xi), wherexi denotesx1, x2, . . . , xn. We also denote

xi =: x.Let i ∈ 1,2, . . . , n. Let m ∈ Z+. We define our first pre-extensionE1(ui) by

Definition 5.

E1(ui) :=

ui(X), if X ∈ DT ,∑m+2k=1 Ak(ui(−t/k, x)), if X ∈ −DT ,

whereAk, k = 1,2, . . . ,m + 2, is the solution of the linear algebraic system:

m+2∑k=1

(−1)sAk = 1, s = 0,1, . . . ,m + 2.

Remark 6. This is the standard extension used to extend functions to reflected domSee, for example, [11, pp. 125–126]. As is shown there, ifui is C0,C1,C2, etc., then


e (i.e.

e

s

ec-

ith

t allsince

n

[10,

E1(ui) is C0, C1, C2, etc., inDT,Double. Note that this extension also preserves anyL2Sobolev space regularity ofui .

Now, we arrange that our extension is compactly supported as a function of timonR). We do this by applying the standard extension operator (denoted byET,2T ), in timeto extendE1(ui) from [−T ,T ] to [−2T ,2T ] [12] with compact support and with the samregularity asE1(ui). Note that this also preserves the spacial regularity.

Definition 7. E(ui) := ET,2T (E1(ui)).

We note thatE(ui) is at least as smooth onR × RN asE1(ui) is onDT,Double.From now on, in this paper, we identifyui with its extensionE(ui), and we suppres

the notationE.

3.2. Sup and Inf convolution regularization

Now, we recall the “quadratic envelope regularization” of [9, p. 44], [10] and [9, Stion 5.1, pp. 43, 44].

Let i ∈ 1,2, . . . ,N. Let ui :R × Rn → R be an upper semicontinuous function wbounded jump. LetH be an open set ofR × Rn such thatH ⊂ R × Rn. Then, we definefor all ε > 0:

Definition 8.

uεi (x0) := sup

x∈H

(ui(x) + ε − 1

ε|x − x0|2

), for x0 ∈ H. (1)

We note that [9]uεi is Lipschitz continuous. It follows by elementary arguments tha

the conclusions of Lemma 5.2, p. 44, and Theorem 5.1(a,b) in [10, p. 43] still hold,ui has finite jump. Note that finite jump is essential for these arguments to obtain.

Definition 9. We define:uε(x0) := uεi (x0), for x0 ∈ H , wherei = 1,2, . . . ,N . In the

same way, leti ∈ 1,2, . . . ,N. Let ui :R × Rn → R be a lower semicontinuous functiowith bounded jump. LetH be as above. Then, we define, for allε > 0:

Definition 10.

ui,ε(x0) := Infx∈H

(ui(x) − ε + 1

ε|x − x0|2

), for x0 ∈ H. (2)

All of the mirror-reflected properties of Lemma 5.2, p. 44, and Theorem 5.1 inp. 43] are still valid, mutis mutandis.

Note thatuεi ↓ ui , for eachi, pointwise (even at the points of discontinuity ofui ), and

ui,ε ↑ ui pointwise (even at the points of discontinuity ofui). This is (a) of Theorem 5.1of [10, p. 43].


th

t is

ts and

at

Geometrically, this is because theuεi and theui,ε are envelopes of parabolas of wid

depending onε, with each parabola above (respectively below), byε, the value ofu at eachpoint of the graph ofu.

Note also, thatuεi andui,ε are Lipschitz continuous, where the Lipschitz constan

proportional to1ε.

Definition 11. We defineuε(x0) := ui,ε(x0), for x0 ∈ H , wherei = 1,2, . . . ,N .

To obtain higher regularity, we Poisson regularize each of theuεi andui,ε .

Let γ > 0, σ > 0.

Definition 12. (uγ

i )σ := (uγ

i ∗ Pσ ), wherePσ is the Poisson kernel.

Definition 13. (ui,γ )σ := (ui,γ ) Pσ , wherePσ is the Poisson kernel.Thus, for eachi ∈ 1,2, . . . ,N , σ > 0, γ > 0, we have (at each point inDT ):

(i)(u

γ

i

)σ

and(ui,γ ) areC∞,

(ii) limγ↓0

[limσ↓0

(u

γ

i

)σ

]= ui,

(iii) limγ↓0

[limσ↓0

(ui,γ )σ

]= ui. (3)

Finally, as we mentioned at the beginning of this section, sinceu is a vector functionwe define:

Definition 14.

(uγ )σ :=

(u

γ

1

)σ(

uγ

2

)σ

...(u

γ

N

)σ

.

Definition 15.

(uγ )σ :=

(u1,γ )σ

(u2,γ )σ...

(uN,γ )σ

.

We will have many occasions in the sequel to compare vector valued constanfunctions. We will use the partial ordering given by

Definition 16. Let u ∈ RN , v ∈ RN , thenu v iff for all i ∈ 1,2, . . . ,N we haveui vi .If c is a real constant the expressionu c denotes (with slight abuse of notation) thu (c, c, . . . , c) ∈ RN . Similarly for u v mutis mutandis.


(and

stem

Let

. 402–em 13

idean

Definition 17. Two vector valued functionsf1 andf2 with values in a subset ofRN satisfyf1 f2 iff f1(X) f2(X) for all X in domain(f1) ∩ domain(f2). Similarly for f1 f2.

4. Semicontinuous (sub-), (super-) and solutions of quasilinear symmetrichyperbolic systems

As in our previous papers [1–3], we define a notion of sub- and supersolutionsolution) for our P.D.E. system by using our technique of regularization.

As in Section 2, Definition 3, we start with a nonlinear, symmetric hyperbolic syL(•) = 0, in a slab domainDT .

Since all the preliminaries are now in place, we state our definition.

Definition 18. Let v :DT → RN be upper semicontinuous, with bounded jump.u :DT → RN be lower semicontinuous, with bounded jump. Then

(a) u is aviscositysubsolution atY ∈ D0T iff:

£−(u)(Y ) := limγ↓0

[limσ↓0

L((uγ )σ

)(Y )

] 0. (4)

(b) v is aviscositysupersolution atY ∈ D0T iff:

£+(v)(Y ) := limγ↓0

[limσ↓0

L((vγ )σ

)(Y )

] 0. (5)

(c) If (a) obtains for allY ∈ D0T , thenu is aviscositysubsolution inD0

T .(d) If (b) obtains for allY ∈ D0

T , thenv is aviscositysupersolution inD0T .

(e) If w :DT → RN satisfies (a) and (b) atY ∈ D0T , thenw is aviscositysolution atY .

(f) If w :DT → RN satisfies (e) at allY ∈ D0T , thenw is aviscositysolution inD0

T .

Remark 19. Note that ifu,v areC2 atY ∈ D0T , then£−(u)(Y ) = L(u)(Y ) and£+(v)(Y ).

5. Comparison principles for C2+ε0 super- and subsolutions

Recall that

L(u) := ∂tu + Ai(t, x,u)∂iu − B(t, x)u − f (t, x) = 0. (6)

We prove our comparison principle by using a parabolic regularization (see [4, pp405]) and then by applying a slight modification of the argument used to prove Theorof [13, p. 190] to the parabolic regularization.

5.1. Notation

Definition 20. Let Ω be a domain in some Euclidean space or cross product of Euclspaces. We denote the Sobolev space in a domainΩ of functions inLp(Ω,RN) withm-derivatives inLp(Ω,RN) by Hm,p(Ω,RN).


for1

-m

roof,e

3,

We denote the closure ofC∞0 (Ω,RN) in Hm,p(Ω,RN) by H 0,m,p(Ω,RN).

Theorem 21 (Comparison principle forC2+ε0 super- and subsolutions). Let L(•) be aquasilinear symmetric first order hyperbolic operator inDT , with Ai and f admissible.Let ε0 ∈ (0,1). Let w1 ∈ C2+ε0(Rn,RN) ∩ H 3,2(Rn,RN) and w2 ∈ C2+ε0(Rn,RN) ∩H 3,2(Rn,RN). Let u1 ∈ c2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) be a subsolution ofL(u1) 0in D0

T , with u1(0, x) = w1(x) for all x ∈ Rn. Let u2 ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN)

be a supersolution ofL(u2) 0 in D0T , withu2(0, x) = w2(x) for all x ∈ Rn. Letw1 w2.

Then,u2 u1 in D0T .

In order to prove Theorem 21 we prove an auxiliary comparison theoremC2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) supersolutions ofL(•) 0. Once proved, Theorem 2will follow immediately. We prove

Theorem 22. LetL(•) be a quasilinear hyperbolic first order operator inDT , withAi andf admissible. Letε0 ∈ (0,1). Letw1 ∈ C2+ε0(Rn,RN) ∩ H 3,2(Rn,RN) with w1 0. Letu1 ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) be a supersolution ofL(u1) 0 in D0

T (under theregularity assumptions of Definition3), with u1(0, x) = w1(x) 0 for all x ∈ Rn. Then,u1 w1 in D0

T .

Proof. Note that inD0T , u1 satisfies:

∂tu1 + Ai(t, x,u1)∂iu1 − B(t, x)u1 − f (t, x) =: g(t, x) 0, (7)

u1(0, x) = w1(x) on Base(DT ), (8)

with g ∈ C1,ε0(D0T ,RN) ∩ H 1,2(DT ,RN).

We now apply parabolic regularization to (7).Let ε > 0. Let∆t be the Laplacian inDT . Let [•]ε be aC∞

0 approximation by convolution with aC∞

0 smoothing kernel with parameterε. We solve the parabolic linear systein D0

T :

∂tuε1 − ε∆tu

ε1 + Ai(t, x,u1)∂iu

ε1 − B(t, x)uε

1 − f (t, x) = g(x, t),

uε1(0, t) = [

wε1(x)

]ε. (9)

First, we will prove a comparison principle for (9), and then at the end of the pwe will show thatuε

1 converges pointwise tou1, which will give our comparison principlfor u1.

5.2. Parabolic comparison

Note that it follows from local and global parabolic regularity [14] thatu1ε is in

C2(DT ,RN) ∩ H 2,2(DT ,RN) (we have used that the initial data is inC2+ε0(Rn,RN) ∩H 3,2(Rn,RN)). Let Ai(t, x,−u1) := Ai(t, x,u1) and that the proof of [13, Theorem 1p. 190], applied to−uε

1, shows that, on any compact rectangle of the formRT :=DT ∩ (t, x) | −R x R, we have that−uε |RT −uε |parabolicboundary(RT ).
1 1


low,

ases de-f

b-measure

t

nd

ndentt:

ument,ons

first

CaseI: Suppose thatuε1(0, x) = 0. But, note that because of the next section just be

uε1 ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) which implies that lim|x|→∞ uε

1 = 0, and we seethat asR → ∞, the maximum of−uε

1 eventually must occur on Base(RT ).CaseII: uε

1(0, x) = 0. In this case, we apply Protter’s theorem again to−uε1 on eachRT ,

and we see that the maximum of−uε1 occurs on the parabolic boundary, and again

R → ∞, we have that maximum cannot exceed smaller and smaller positive valupending onR, because lim|x|→∞ uε

1 = 0. Hence, onDT , we have that the maximum o−uε

1 = 0. Thus, in this case,−uε1(t, x) −uε

1(0, x) for all (t, x) ∈ DT .Now, we show that a subsequence of the parabolic regularizationsuε

1 converge point-wise tou1.

It follows from [4, Theorems 1–4, pp. 402–408] that a subsequence ofuε1 converges

to u1 weakly in Sobolev sense and strongly inL2 sense. However, we show that a susequence of this sequence can be chosen (after the usual modification on sets ofzero) to converge in sup norm everywhere tou1.

Fix ε > 0. Recall [14, Theorem 3.1, p. 582] that for any(n + 1)-rectangular compacsubdomainRT of DT , with finite distanced(RT , ∂DT ), we have, for someα1 ∈ (0,1),that uε

1 ∈ Cα1,α1/2(RT ) with (α,α/2)-Hölder norm bounded above by a universal bouindependent ofε, but depending ond(RT ,DT ) and on essmaxRT

(uε1). Also, recall [14,

Theorem 4.1, p. 584], that: inRT , for someα2 ∈ (0,1), all first partials ofuε1 are in

Cα2,α2/2(RT ), again with Hölder norm bounded above by a universal bound indepeof ε, but depending ond(RT ,DT ) and‖uε

1‖L2. Now, recall [14, Theorem 2.2, p. 579] thaby ess maxRT

(uε1) is bounded above by the (L2 spaceL∞ time) norm ofuε

1 in DT , which[4, pp. 402–408] is bounded universally independently ofε, in DT .

Combining the above, we haveC1,α,α/2(RT ) bounds onuε1, for someα > 0, indepen-

dent ofε (depending ond(RT ,DT )). We take a countable covering ofDT by such (finitelyoverlapping) rectangles, and apply the Arzela–Ascoli theorem and a diagonal argto obtain a subsequence of Evans’ sequence that converges to a limit vector functiv intheC1,α,α/2 topology of compact convergence inDT . A fortiori, this sequence convergepointwise tov. Hence, Evans’ limit function is actually aC1

loc(D0T ) solution of (6).

But, v has the same continuous initial data asu1, and they are bothC1loc solutions of (6)

in any compact subdomain ofD0T . Such solutions are unique (uniqueness theorem for

order quasilinear symmetric hyperbolic systems [5, p. 22]). Hencev = u1.Finally, to complete the proof of Theorem 22, we have thatu1(t, x) = v(t, x) ← uε

1(t, x)

(along this subsequence) wε1(t, x) → w1(t, x) asε ↓ 0, for all (t, x) ∈ D0

T . Now, we consideru2, as in the statement of Theorem 21. Note that, inDT , u2 satisfies:

∂tu2 + Ai(t, x,u2)∂iu2 − B(t, x)u2 − f (t, x) =: h2(t, x) 0,

∂tu2 + Ai(t, x,u2)∂iu2 − B(t, x)u2 − f (t, x) =: h2(t, x) 0. (10)

We prove Theorem 21, using Theorem 22.Consideru := u1 − u2. Then,u satisfies

∂tu + Ai(t, x,u1)∂iu1 + Ai(t, x,u1)∂iu2 − Ai(t, x,u1)∂iu2

− Ai(t, x,u2)∂iu2 − B(t, x)u 0. (11)


theo-

ions inorison

s

xists

for aprooferties

s

We define˜

Ai(t, x,u) by

Ãi(t, x,u)∂iu := Ai(t, x,u1)∂iu1 − Ai(t, x,u1)∂iu2. (12)

Note that˜

Ai is a symmetricC∞ function of its arguments,

ut + Ãi(t, x,u)∂iu − B(t, x)u f0(t, x) := [

Ai(t, x,u2) − Ai(t, x,u1)]∂iu2, (13)

ut + Ãi(t, x,u)u − B(t, x)u f0(t, x) := [

Ai(t, x,u2) − Ai(t, x,u1)]∂iu2,

u(0, x) = 0. (14)

System I is of the form treated with the hypothesis of Theorem 22. Applying thisrem, we obtain thatu = u1−u2 0 in DT .

6. Comparison principle for semicontinuous viscosity super- and subsolutions

In the previous section, we proved a comparison principle for super- and subsolutthe classC2+ε0(DT ,RN)∩H 2,2(DT ,RN). In this section, we prove a similar principle fsemicontinuous viscosity solutions. It will follow by approximation, using the comparprinciple of the previous section.

Theorem 23. Let ε0 ∈ (0,1). Letw ∈ C2+ε0(Rn,RN) ∩ H 2,2(Rn,RN). Letv be an uppersemicontinuous function fromDT to RN , with bounded jump, and letv(0, x) = w(x).Let v be a viscosity supersolution of £+(v) 0 in D0

T . Let u be a lower semicontinuoufunction fromDT to RN , with bounded jump, and letu(0, x) = w(x). Letu be a viscositysubsolution of £−(u) 0 in D0

T . Then,v u in DT .

Proof. If not, then there exists an interior pointP0 = (t0, x0) with u(P0) − v(P ) > ε > 0.It follows from two Theorems 15 and 20 of Appendices A and B (Part II), that there ea subsolutionUε of L(Uε) 0, in D0

T , with Uε ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) anda supersolutionVε of L(Vε) 0 in D0

T with Vε ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) suchthatUε(0, x) = Vε(0, x) = w(x) and such thatV (P0) < Vε(P0) < Uε(P0) < U(P0). But,this contradicts the comparison theorem of the previous section.

7. A difference criterion for viscosity subsolutions

In this section, similarly to [3, Section 2, p. 558], we provide a difference criterionlower semicontinuous function with bounded jump to be a viscosity subsolution. Ouris a vector generalization of that in [3], but is simplified somewhat by the magic propof sup convolution.

Theorem 24. Let DT be a slab domain. Letε0 ∈ (0,1). Let u be a lower semicontinuoufunctionu :DT → RN with bounded jump. Letw ∈ C2+ε0(Rn,RN) ∩ H 3,2(Rn,RN) and


ced-

-

s

let u(0, x) = w(x). Then,u is a viscosity subsolution of £−(u) 0 in D0T iff ϕ − u 0

∀ϕ ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) such thatL(ϕ) 0 in D0T andϕ(0, x) = w(x).

Proof. (⇒) Under the hypothesis, it follows from the comparison theorem of the preing section, Theorem 23, thatu ϕ in DT .

(⇐) We prove the contrapositive of(⇐). Supposeu is not a viscosity subsolution of £−(u) 0. Then, there existsY0 = (t0, y0) ∈ DT and ∃j ∈ 1,2, . . . ,N with[£−(u)]j (Y0) = c2

1 > 0.Case1: First, we assume that[£−(u)]i (Y0) = −∞ for i = j .We define an auxiliary functionm. Let µ0 > 0 be such thatt0 + µ0 < T . Let

0< µ µ0,

m(t0,µ, s,ω)(t) :=

s for t0 + µ t T ,

g(t) for t0 t t0 + µ,

0 for 0 t t0,

(15)

whereg : [t0,t0 + µ] → R+ is aC∞ function with 0< g(t) < s, and with dg(t)dt

|t=t0 = ω.Note,g is a function oft , that “ramps” up from 0 att = t0 to a smalls > 0, in small

timeµ with large positive derivativeω at t = t0.Then, there existβj ∈ R+ ands,µ0,ω, k1, k2, k3 such that

Ψ (t, x) :=

Ψ i(t, x) := ui(t, x) + m(t0,µ, s,ω)(t)e−k21(t−t0)

2e−k2

2(x−y0)2,

Ψ j (t, x) := uj (t, x) − βj te−k23(t−t0)

2e−k2

4(x−y0)2,

(16)

satisfies[£−(Ψ )(Y0)]i > c20 > 0, as well as[£−(Ψ )(Y0)]j > c2

0 > 0 (for somec0); andΨ isbounded with bounded jump inDT . Note thatΨ (0, x) = w(x). Note thatΨ j (Y0) < uj (Y0)

andΨ j (Y0) − uj (Y0) = O(β).We have used theC1-ness of the coefficients ofL, the fact that 0< t < T < ∞, the

exponential weighting factors inΨ , and the fact thatu is bounded atY0. Note thatωdepends on theβj .

Note thatui(Y0) = Ψ i(Y0) for i = j . At no loss of generality,s > 0 can be chosen alarge as we wish.

Now, we apply Theorem 15 of Appendix A, and we obtain (for vectorε > 0 smallenough) a functionϕ satisfying(i = 1,2, . . . ,N):

(i) ϕ ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN),

(ii) L(ϕ) c24 in DT ,

(iii) ϕ(0, x) = w(x),

(iv) Ψ (Y0) < ϕ(Y0) Ψ (Y0) + ε,

(v) ui(Y0) = Ψ0(Y0) < Ψ i(Y0) + ε = ui(Y0) + ε, i = j,

(vi) uj (Y0) > Ψ j (Y0),

(vii) uj (Y0) − Ψ j (Y0) = O(β). (17)


i-

eo-ase 1

con-

-

of the

2002)

2002)

2003)

Now chooseε > 0 small enough, andk5, k6 so that, givenϕ :DT → RN defined by

ϕ := ϕ − δte−k25(t−t0)

2e−k2

6(x−y0)2 ∃δ > 0, (18)

satisfiesϕi (Y0) < ui(Y0) and ϕ satisfies (i)–(iv) above withc24 replaced by some pos

tive c25.

Then ϕ is a supersolution inD0T with the same initial data asu, in C2+ε0(DT ,RN) ∩

H 2,2(DT ,RN) and[ϕ − u]j < 0. We have shown the contrapositive of(⇐).We now handle the second case.Case2: Suppose that∃i ∈ 1,2, . . . ,N, [£−(u)]i (Y0) = −∞. Note that for anyε > 0

we can chooseγ (ε), σ (ε) > 0 such that

(a) ui(Y0) [ui]γ (ε)

σ (ε)(Y0) ui(Y0) + ε,

(b)[L

((u)

γ (ε)

σ (ε)

)]i(Y0) > −∞. (19)

Now notice that, after definingΨ as above, and using (a), (b) in the proof of Threms 15 and 20 of Appendices A and B, these theorems still hold, and the proof of Cabove is now obtained, mutis mutandis.

8. Difference criterion for viscosity supersolutions

As in [3, p. 562] , we have a “mirror image” difference criterion for an upper semitinuous function, fromDT to RN , with bounded jump, to be a viscosity supersolution.

Theorem 25. Let ε0 ∈ (0,1). Let DT be a slab domain. Letv be an upper semicontinuous functionv :DT → RN with bounded jump. Letε0 ∈ (0,1). Let w ∈ C2+ε0(Rn) ∩H 3,2(RN). Let v(0, x) = w(x). Then,v is a viscosity solution of £+(v) 0 in D0

T iff∀ϕ ∈ C2+ε0(DT ,RN) ∩ H 2,2(DT ,RN) such thatL(ϕ) 0 andϕ(0, x) = w(x), we havev − ϕ 0 in DT .

Proof. Same as that of Theorem 24 mutis mutandis. The proof is a mirror imageproof of that theorem.

This completes Part I of this paper.

References

[1] P. Smith, Perron’s method for semilinear hyperbolic equations: Part I, J. Math. Anal. Appl. 275 (693–710.

[2] P. Smith, Perron’s method for semilinear hyperbolic equations: Part II, J. Math. Anal. Appl. 276 (628–641.

[3] P. Smith, Perron’s method for semilinear hyperbolic equations: Part III, J. Math. Anal. Appl. 283 (557–569.

[4] L.C. Evans, Partial Differential Equations, American Mathematical Society, 1991.[5] O. Reula, Hyperbolic methods for Einstein’s equations, http://relativity.livingreviews.org/lrr-1998-3.


ms, in:

th. 48

Type,

[6] J.E. Marsden, A.E. Fischer, General relativity, partial differential equations, and dynamical systePartial Differential Equations, in: Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., 1974.

[7] D.M. Deturk, The Cauchy problem for Lorentz metrics with prescribed Ricci curvature, Compos. Ma(1983) 327–349.

[8] R. Courant, D. Hilbert, Methods of Mathematical Physics, Wiley–Interscience, 1962.[9] F.H. Clarke, et al., Nonsmooth Analysis and Control Theory, Springer, 1991.

[10] L. Caffarelli, X. Cabre, Fully Nonlinear Equations, Amer. Math. Soc., 1991.[11] L. Mikhailov, Partial Differential Equations, Mir, 1978.[12] R.A. Adams, Sobolev Spaces, Academic Press, 1995.[13] M. Protter, H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, 1967.[14] V.A. Ladyzenskya, V.A. Solenikov, N.N. Uraltzeva, Linear and Quasilinear Equations of Parabolic

Amer. Math. Soc., 1968.

Further reading

[15] N.L. Carothers, Real Analysis, Cambridge Univ. Press, 1999.[16] M. Taylor, Partial Differential Equations, vol. 1, Springer, 1996.[17] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, 1965.[18] C.L. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.

Perron's method for quasilinear hyperbolic systems: Part I

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