Handbook of differential equations : stationary partial differential equations

595 Pages·2008·25.458 MB·English

by  M Chipot

Preface
This handbook is the sixth and last volume in the series devoted to stationary partial
differentialequations. As theprecedingvolumes, itisacollectionofselfcontained, state
of-the-artsurveyswrittenbywell-knownexpertsinthefield.
The topics covered by this volume include in particular domain perturbations for
boundary value problems, singular solutions of semi-linear elliptic problems, positive
solutions to elliptic equations on unbounded domains, symmetry ofsolutions, stationary
compressible Navier-Stokes equation, Lotka-Volterrasystems withcross-diffusion, fixed
pointtheoryforelliptic boundaryvalueproblems. Ihope thatthese surveyswill be useful
forbothbeginnersandexpertsandhelptoawidediffusionoftheserecentanddeepresults
inmathematicalscience.
I wouldlike to thank all the contributors for theirelegantarticles. I also thankLauren
Schultz Yuhasz andMaraVos-SarmientoatElsevierfor the excellentediting workofthis
volume.
M.Chipot
v
List of Contributors
Daners, D., SchoolofMathematics and Statistics, The University ofSydney, NSW2006,
[email protected] (Ch. 1)
Davila, J., Departamento de Ingenierfa Matenuitica and CMM, Universidad de Chile,
Casilla 170Correo 3, Santiago, [email protected](Ch. 2)
v.,
Kondratiev, Department ofMathematics andMechanics, Moscow State University,
Moscow 119899, [email protected] (Ch.3)
v.,
Liskevich, Department ofMathematics, University ofSwansea, Swansea SA2 8Pp,
[email protected](Ch. 3)
Sobol, Z., Department ofMathematics, University ofSwansea, Swansea SA2 8Pp, UK
[email protected](Ch. 3)
Pacella, F., Dipartimento diMatematica, Universita di Roma La Sapienza, P.leA. Moro
2, 00185, Roma, [email protected](Ch. 4)
Ramaswamy, M., TIFR-CAM, Yelahanka New Town, Bangalore 560065, India
[email protected] (Ch. 4)
Plotnikov, P.I., Lavryentyev Institute of Hydrodynamics, Siberian Division of
Russian Academy of Sciences, Lavryentyev pI: 15, Novosibirsk 630090, Russia
[email protected] (Ch. 5)
Sokolowski, J., Institut Elie CaI·tan, Laboratoire de Mathematiques, Universite Henri
Poincare, Nancy 1, B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France
[email protected] (Ch.5)
Yamada, Y., Department ofApplied Mathematics, Waseda University, 3-4-1 Ohkubo,
Shinjuku-ku, Tokyo 169-8555, Japan [email protected] (Ch.6)
Zon, H., Department of Mathematics, University ofAlabama at Birmingham, USA
[email protected] (Ch.7).
vii
CHAPlER 1
Domain Perturbation for Linear and Semi-Linear
Boundary Value Problems
Daniel Daners
SchoolofMathell/£lticsandStatistics. TheUniversityofSydney.NSW2006.Australia
E-/Iutil:D.Daners@/I1£1ths.usyd.edu.au
URL:www./I1£1ths.usyd.edu.au/u/daners/
Abstract
This is a survey on elliptic boundary value problems on varying domains and tools
neededforthat. Suchproblemsariseinnumericalanalysis,inshapeoptimisationprob
lems and in the investigation ofthe solution structure ofnonlinearelliptic equations.
Themethodsare also usefultoobtaincertainresultsfor equationsonnon-smoothdo
mainsbyapproximationbysmoothdomains.
Domain independentestimates and smoothing properties are an essential tool to
deal with domain perturbation problems, especially for non-linear equations. Hence
we discuss such estimates extensively, together with some abstract results on linear
operators.
Asecondmajorpartdealswithspecificdomainperturbationresultsforlinearequa
tions with various boundaryconditions. We completelycharacterise convergence for
Dirichlet boundary conditions and also give simple sufficient conditions. We then
prove boundary homogenisation results for Robin boundary conditions on domains
with fast oscillating boundaries, where the boundary condition changes in the limit.
We finallymentionsome simpleresults onproblemswithNeumann boundarycondi
tions.
The final partis concernedaboutnon-linearproblems, using the Leray-Schauder
degree to prove the existence of solutions on slightly perturbed domains. We also
demonstrate how to use the approximation results to getsolutions to nonlinearequa
tionsonunboundeddomains.
Keywords: Elliptic boundaryvalue problem, Domainperturbation, Semilinearequa
tions,Aprioriestimates,Boundaryhomogenisation
HANDBOOKOFDIFFERENTIALEQUATIONS
StationaryPartialDifferentialEquations.volume6
EditedbyM.Chipot
©2008ElsevierB.V.Allrightsreserved
2 D.Dallers
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Ellipticboundaryvalueproblemsindivergenceform. . . 5
2.1. Weaksolutionstoellipticboundaryvalueproblems. 5
2.2. Abstractformulationofboundaryvalueproblems. 9
2.3. Formallyadjointproblems . 12
2.4. Globalaprioriestimatesforweaksolutions . . . . . 13
2.4.1. SobolevinequalitiesassociatedwithDirichletproblems . 16
2.4.2. Maz'ya'sinequalityandRobinproblems . 17
2.4.3. SobolevinequalitiesassociatedwithNeumannproblems. 18
2.5. Thepseudo-resolventassociatedwithboundaryvalueproblems 18
3. Semi-linearellipticproblems . 20
3.1. Abstractformulationofsemi-linearproblems 20
3.2. Boundednessofweaksolutions .. 22
4. Abstractresultsonlinearoperators . . . 23
4.1. Convergenceintheoperatornorm . 24
4.2. Aspectralmappingtheorem . . . . 25
4.3. Convergencepropertiesofresolventandspectrum. 26
5. PerturbationsforlinearDirichletproblems 29
5.1. Assumptionsandpreliminaryresults. 29
5.2. Themainconvergenceresult..... 31
5.3. Necessaryconditionsforconvergence 35
5.4. Sufficientconditionsforconvergence 37
5.5. Proofofthemainconvergenceresult. 39
6. VaryingdomainsandRobinboundaryconditions. 42
6.1. Summaryofresults. . . . . . . . . . . . . . 42
6.2. Preliminaryresults . . . . . . . . . . . . . . 45
6.3. Smallmodificationsoftheoriginalboundary 47
6.4. Boundaryhomogenisation: LimitisaDirichletproblem 50
6.5. Boundaryhomogenisation: LimitisaRobinproblem. 53
7. Neumannproblemsonvaryingdomains . 55
7.1. RemarksonNeumannproblems . 55
7.2. ConvergenceresultsforNeumannproblems. 56
8. Approximationbysmoothdataanddomains . . . 58
8.1. Approximationbyoperatorshavingsmoothcoefficients. 58
8.2. Approximationbysmoothdomainsfromtheinterior . . 60
8.3. ApproximationfromtheexteriorforLipschitzdomains. 60
9. Perturbationofsemi-linearproblems . 62
9.1. Basicconvergenceresultsforsemi-linearproblems .. 62
9.2. Existenceofnearbysolutionsforsemi-linearproblems 65
9.3. Applicationstoboundaryvalueproblems 68
9.4. Remarksonlargesolutions..... 70
9.5. Solutionsbydomainapproximation 72
9.6. Problemsonunboundeddomains 73
References. . . . . . . . . . . . . . . . . . 75
DOII/ninperturbationforlinearandsell/i-linearboundaryvalueproblems 3
1. Introduction
Thepurposeofthis surveyis tolookatellipticboundaryvalueproblems
Anu= f in Qn,
Hnu=O oniJQn
withallmajortypesofboundaryconditionsonasequenceofopensets Qn inJRN (N ~ 2).
We then study conditions under which the solutions converge to a solution of a limit
problem
Au= f in Q,
Hu = 0 on iJQ
on some open set Q C JRN. In the simplest case A,An = - ~ is the negative Laplace
operator, and Hn•H the Dirichlet, Robin or Neumann boundary operator, but we work
with general non-selfadjoint elliptic operators in divergence form. We are interested in
verysingularperturbation,notnecessarilyofatype suchthatachangeofvariablescan be
appliedtoreducetheproblemontoafixeddomain. Forthetheoryofsmoothperturbations
rathercomplementarytoours wereferto [84] andreferences therein.
Themainfeatures ofthis expositionare the following:
• We presentan Lp-theoryoflinearand semi-linearelliptic boundary value problems
withdomainperturbationinview.
• We establish domain perturbation results for linear elliptic problems with Dirichlet,
RobinandNeumannboundaryconditions, applicable tosemi-linearproblems.
• We showhow to use the linearperturbationtheory to deal with semi-linearproblems
on bounded and unbounded domains. In particular we show how to get multiple
solutions for simple equations, discuss the issue of precise multiplicity and the
occurrenceoflargesolutions.
• We provide abstract perturbation theorems useful also for perturbations other than
domainperturbations.
• We provide tools to prove results for linear and nonlinear equations on general
domains bymeans ofsmoothingdomainsandoperators (see Section8).
Ouraimis tobuildadomainperturbationtheorysuitableforapplications tosemi-linear
problems,thatis,problemswheref = f (x. u(x))isafunctionofx aswellasthesolution
u(x). Fornonlinearities with growth, polynomial orarbitrary, we needa good theory for
the linearproblem in Lp for 1 < P < 00. Good in the contextofdomain perturbations
meansthatinallestimatesthereiscontrolondomain dependenceoftheconstantsinvolved.
WeestablishsuchatheoryinSection2.1,wherewealsointroducepreciseassumptionson
the operators. Startingfrom adefinitionofweaksolutions weprove smoothing properties
ofthe corresponding resolvent operators with control on domain dependence. The main
results are Theorems 2.4.1 and 2.4.2. In particular we prove that the resolventoperators
have smoothing properties independent ofthe domain for Dirichlet and Robin boundary
conditions, butnot for Neumann boundary conditions. To be able to work in a common
spaceweconsidertheresolventoperatorasamapactingon Lp(JRN), sothatitbecomesa
pseudo-resolvent(see Section2.5).
4 D.Dallers
The smoothing properties ofthe resolvent operators enable us to reformulate a semi
linear boundary value problem as a fixed point problem in Lp(JRN). Which P E (1, 00)
wechoosedependsonthegrowthofthenonlinearity. Theruleis, thefasterthegrowth, the
larger the choice ofp. We also show that under suitable growth conditions, a solution in
Lp is infact in Loo. Again, the focus is ongetting controloverthe domain dependenceof
the constantsinvolved. Forapreciseformulation oftheseresults werefertoSection3.
Let the resolvent operator corresponding to the linear problems be denoted by RIlCA)
and RCA). The key to be able to pass from perturbations ofthe linearto perturbations of
thenonlinearproblemis thefollowing propertyoftheresolvents:
If!Il ~ ! weakly, thenRIlCA)!1l---+ ROc)!strongly.
Henceforalltypes ofboundaryconditionsweprovesuchastatement. IfR(A) iscompact,
it turns out that the above property is equivalent to convergence in the operator norm.
An issue connected with that is also the convergence ofthe spectrum. We show that the
abovepropertyimpliesthe convergenceofeveryfinite partofthe spectrumoftherelevant
differential operators. We also show here that itis sufficientto prove convergence ofthe
resolvent in Lp(JRN) for some p and some Ato have them for all. These abstractresults
are collectedinSection4.
ThemostcompleteconvergenceresultsareknownfortheDirichletproblem(Section5).
Thelimitproblemis always aDirichletproblemonsomedomain. Weextensivelydiscuss
convergence in the operator norm. In particular, we look at necessary and sufficient
conditions for pointwise and uniform convergence of the resolvent operators. As a
corollary to the characterisation we see that convergence is independent ofthe operator
chosen. Wealsogivesimplesufficientconditionsforconvergenceinterms ofpropertiesof
the setQIl n Qc. Themainsourcefor theseresults is [58].
ThesituationisrathermorecomplicatedforRobinboundaryconditions,where thetype
of boundary condition can change in the limit problem. In Section 6 we present three
differentcases. First, we lookatproblems with only a smallperturbation ofiJQ. We can
cutholes and add thin pieces outside Q, connectedto Q onlyneara setofcapacity zero.
Second, we look at approximating domains with very fast oscillating boundary. In that
case the limit problem has Dirichlet boundary conditions. Third, we deal with domains
withoscillatingboundary, suchthatthelimitprobleminvolvesRobinboundaryconditions
withadifferentweightonthe boundary. The secondandthirdresults are really boundary
homogenisationresults. Theseresults arealltakenfrom [51].
The Neumann problem is very badly behaved, and without quite severe restriction on
the sequence ofdomains QIl we cannot expect the resolvent to converge in the operator
norm. Inparticularthespectrumdoesnotconverge,asalreadynotedin [40,page420]. We
only prove a simple convergence resultfitting into the general framework establishedfor
the otherboundaryconditions.
Afterdealing withlinearequations weconsidersemi-linearequations. Alotofthispart
is inspiredbyDancer'spaper [45] andrelatedwork. Theapproachis quite differentsince
wetreatlinearequationsfirst, andthenusetheirpropertiestodealwithnonlinearequations.
Theideaistousedegreetheorytogetsolutionsonanearbydomain, givenasolutionofthe
limitproblem. Thecoreoftheargumentisanabstracttopologicalargumentwhichmaybe
usefulalsofor othertypes ofperturbations (see Section9.2). Wealso discuss the issue of
DOII/ninperturbationforlinearandsell/i-linearboundaryvalueproblems 5
precisemultiplicityofsolutions andthe phenomenonoflarge solutions. Finally, we show
thatthe theoryalsoapplies tounboundedlimitdomains.
There are many other motivations to look at domain perturbation problems, so for
instance variational inequalities (see [102]), numerical analysis (see [77,107,110,116
119]), potential and scattering theory (see [10,108,113,124]), control and optimisation
(see [31,34,82,120]), r-convergence (see [24,42]) and solution structures of nonlinear
elliptic equations (see [45,47,52,69]). We mention more references in the discussion on
the specific boundary conditions. Someresults go backalong time, see for instance [19]
or [40]. The techniques are even older for the Dirichletproblem for harmonic functions
withthepioneering work [93].
Finally, therearemanyresults wedonotevenmention, sofor instancefor convergence
inthe Loo-normwereferto [8,9,14,23,26]. Furthermore, similarresults can beprovedfor
parabolic problems. The key for thatare domain-independentheatkernel estimates. See
for instance [7,17,47,52,59,78] and references therein. The above is only a small rather
arbitrary selectionofreferences.
2. Ellipticboundaryvalueproblemsindivergenceform
The purpose of this section is to give a summary ofresults on elliptic boundary value
problems in divergence form with emphasis on estimates with control over the domain
dependence.
2.1. Weaksolutions to ellipticboundaryvalueproblems
Weconsiderboundaryvalueproblemsoftheform
Au = f in Q,
(2.1.1)
Bu = 0 on iJQ
on an open subset ofJRN, not necessarily bounded or connected. Here A is an elliptic
operatorindivergenceformandB aboundaryoperatortobe specifiedlaterinthis section.
TheoperatorA is oftheform
-div(AoV'u +au) +b· V'u +cou (2.1.2)
with Ao E Loo(Q. JRN'-;N), a. b E Loo(Q. JRN) and Co E Loo(Q). Moreover, we assume
that Ao(x)is positive definite, uniformly with respect to x E Q. More precisely, there
exists aconstantao > 0suchthat
aol~12 :s ~TAo(x)~ (2.1.3)
forall~ E JRN andalmostallx E Q. Wecallao the ellipticityconstant.
REMARK 2.1.1. We only defined the operator A on Q, but we can extend it to JRN by
setting a = b = 0, Co = 0 and A(x) := aoI on QC. Then the extendedoperator A also
6 D.Dallers
satisfies (2.1.3). In particularthe ellipticityproperty (2.1.3) holds. Hence withoutloss of
generalitywecauassume thatA isdefinedonJRN.
Wefurtherdefine the co-normalderivativeassociatedwith A on aQ by
au
+
-- := (Ao(x)Vu a(x)u) .v,
aVA
where vis theoutwardpointing unitnormalto aQ. AssumingthataQis thedisjointunion
offl, f2 and f3 wedefine the boundaryoperatorE by
ulrl on f I (Dirichletb.c.),
au
Eu := on f2 (Neumann b.c.), (2.1.4)
aVA
au
-- +bou on f3 (Robinb.c.)
aVA
with bo E Leo(f3) nonnegative. Ifallfunctions involvedare sufficiently smooth, then by
theproductrule
+ + +
-vdiv(AoVu au) = (AoVu au) .Vv - div(v(AoVu au))
andtherefore, ifQ admits the divergence theorem, then
-lVdiV(Aovu+aU)dX
= J{Q(AoVu+au).Vvdx- J{aQ(v(AoVu+au))·vdC5,
where C5 is the surfacemeasure on aQ. Hence, ifu is sufficiently smoothand v E ClUJ)
with v = 0on fl, then
{vAudx= {(AoVu+au).Vv+(a.Vu+cou)vdx+ (bouvdC5.
JQ JQ Jr3
The expression on the right-hand side defines a bilinear form. We denote by HI(Q) the
usual Sobolev space ofsquare integrable functions having square integrable weak partial
derivatives. Moreover, H6(Q) is the closureofthe setoftestfunctions C~(Q) in HI(Q).
DEFINITION2.1.2. Foru, v E HI(Q) weset
ao(zt,v):= l(Aovu+aU),vv+(b,VU+COU)VdX.
Theexpression
+ {
a(u, v) :=ao(zt, v) bouvdC5
J
r
3
iscalledthe bilinearformassociatedwith (A,E).
DOII/ninperturbationforlinearandsell/i-linearboundaryvalueproblems 7
Ifuisasufficientlysmoothsolutionof(2.1.1), then
l
a(u. v) = (I. v) := jvdx (2.1.5)
forallv E CI(Q)with v = 0onfl. Notethat(2.1.5)doesnotjustmakesenseforclassical
solutions of(2.1.1), but for u E HI(Q) as long as the boundary integral is defined. We
therefore generalise the notionofsolutionandjustrequire that u is in asuitable subspace
V ofHI(Q) and(2.1.5)forall v E V.
ASSUMPTION 2.1.3. Werequire that V beaHilbertspace suchthat V is dense in L2(Q),
that
andthat
[3
{u 2
E CI(Q): suppu c Q \ fl. bolul dO' < oo} C V.
Iff3 isnonsmoothwereplacethesurfacemeasure0' bythe(N-1)-dimensionalHausdorff
measure sothatthe boundaryintegralmakes sense. Wealsorequire that
(2.1.6)
isanequivalentnormon V.
Wenextconsidersome specific specialcases.
EXAMPLE 2.1.4. (a) For a homogeneous Dirichlet problem we assume that fl = aQ
and let V := H6(Q). For the norm we can choose the usual HI-norm,but on bounded
domains we couldjust use the equivalent norm II'YuI12. More generally, on domains Q
lying between two hyperplanes of distance D, we can work with the equivalent norm
II'Yu112 becauseofFriedrich's inequality
(2.1.7)
validforall u E H6(Q) (see [1l1, TheoremII.2.D]).
(b) For a homogeneous Neumann problem we assume that f2 = aQ and let V :=
HI(Q) with the usualnorm.
(c)Forahomogeneous Robinproblemwe assume thatf3 = aQ. In this expositionwe
will always assume that Q is a Lipschitz domain when working with Robin boundary
conditions and choose V := HI(Q). On a bounded domain we can work with the
equivalentnorm
(2.1.8)
(see [1l1, TheoremIII.5.C] or [56]). Itis possibletoadmitarbitrarydomains as shownin
[11,56].
Wefinally define whatwemeanbyaweaksolutionof(2.1.1).