Table Of ContentDe Gruyter Series in Nonlinear Analysis
and Applications 21
EditorinChief
JürgenAppell,Würzburg,Germany
Editors
CatherineBandle,Basel,Switzerland
AlainBensoussan,Richardson,Texas,USA
AvnerFriedman,Columbus,Ohio,USA
Karl-HeinzHoffmann,Munich,Germany
MikioKato,Kitakyushu,Japan
UmbertoMosco,Worcester,Massachusetts,USA
LouisNirenberg,NewYork,USA
BorisN.Sadovsky,Voronezh,Russia
AlfonsoVignoli,Rome,Italy
KatrinWendland,Freiburg, Germany
Moshe Marcus
Laurent Véron
Nonlinear Second
Order Elliptic
Equations Involving
Measures
De Gruyter
MathematicsSubjectClassification2010:Primary:35-02,35J61,35R06,35J25,35J91;
Secondary:28A33,31A05,46E35.
Authors
Prof.Dr.MosheMarcus
Technion–IsraelInstituteofTechnology
Dept.ofMathematics
TechnionCity
32000Haifa
Israel
[email protected]
Prof.Dr.LaurentVéron
LaboratoiredeMathematiques
CNRSUMR6083
FacultedesSciencesetTechniques
UniversiteFrancoisRabelais
ParcdeGrandmont
37200Tours
France
[email protected]
ISBN978-3-11-030515-9
e-ISBN978-3-11-030531-9
Set-ISBN978-3-11-030532-6
ISSN0941-813X
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Preface
In the last 40 years semilinear elliptic equations became a central subject of study
inthetheoryofnonlinearpartialdifferential equations.Ontheonehand,theinterest
in this area is of a theoretical nature, due to its deep relations to other branches of
mathematics, especially linear and nonlinear harmonic analysis and probability. On
the other hand,this studyis of interest because of its applications. Equationsof this
typecomeupinvariousareassuchas:problemsofphysicsandastrophysics,problems
of differential geometry, logisticproblems related for instance to populationmodels
and,mostimportantly,thestudyofbranchingprocessesandsuperdiffusions.
An important family of such equations is that involving an absorption term, the
modelof which is (cid:2)(cid:2)uCg.x,u/ D 0 where ug.x,u/ (cid:3) 0. Suchequationsare of
particular interest because inthem we havetwo competingeffects: the diffusionex-
pressedbythelineardifferentialpartandtheabsorptionproducedbythenonlinearity
g.Furthermore,equationsofthistypewithpowernonlinearitiesplayacrucialrolein
thestudyofsuperdiffusions.
Naturally,thestudyofsemilinearproblemsisbasedonlineartheoryandinpartic-
ularonthetheoryofboundaryvalueproblemswithL1and,moregenerally,measure
data. In addition to the classical theory of the Laplace equation, this study requires
certainideasofharmonicanalysissuchastheHerglotztheoremonboundarytraceof
positive harmonic functions and the resulting integral representation, Kato’s lemma
and the boundaryHarnack principle. These topics and their applicationto boundary
valueproblemsaretreatedinthefirstchapter.
Inthesecondchapterweturntothemaintopicofthismonograph:boundaryvalue
problemsforthesemilinearproblem
(cid:2)(cid:2)uCg.x,u/Df in(cid:3)
(1)
uDh on@(cid:3)
wheref andhareL1functionsormoregenerallymeasures.Generallyweassumethat
t 7!g.(cid:4),t/isacontinuousmappingfromRintoL1.(cid:3);(cid:4)/,where(cid:4).x/Ddist.x,@(cid:3)/,
thatg.x,(cid:4)/isnon-decreasingforeveryx 2(cid:3)andg.x,0/D0.(L1.(cid:3);(cid:4)/denotesthe
weightedLebesguespacewithweight(cid:4).)Inadditionweassumethat
lim g.(cid:4),t/=t D 1 (2)
t!1
uniformlywithrespecttox incompactsubsetsof(cid:3).Twostandardexamples:
g.x,t/D(cid:4).x/ˇjtjq(cid:2)1t, g.x,t/D expt (cid:2)1. (3)
vi Preface
The problem (1) is understood in a weak sense; we require that u 2 L1.(cid:3)/ and
g ıu 2 L1.(cid:3);(cid:4)/, that the equationholdsin the distributionsense andthat the data
isattainedinaweaksense,relatedtoweakconvergenceofmeasures.Inadditionitis
assumedthatf 2 L1.(cid:3);(cid:4)/or,moregenerally,f D(cid:5) 2 M.(cid:3);(cid:4)/,i.e.,(cid:5)isaBorel
measurein(cid:3)suchthat Z
(cid:4)dj(cid:5)j<1.
(cid:2)
For the boundarydata, it is assumed that h 2 L1.@(cid:3)/ or, more generally, h D (cid:6) 2
M.@(cid:3)/,i.e.,(cid:6) isafiniteBorelmeasureon@(cid:3).
ProblemswithL1dataarediscussedinSection2.1.Inthiscasetheboundaryvalue
problempossessesauniquesolutionu2 L1.(cid:3)/suchthatgıu2 L1.(cid:3);(cid:4)/forevery
f 2 L1.(cid:3);(cid:4)/andh2L1.@(cid:3)/.
An interesting feature of boundary value problems with measure data is that, in
general,theproblemisnotsolvableforeverymeasure.If(1)hasasolutionforhD0
andameasure f D (cid:5) 2 M.(cid:3);(cid:4)/,we saythat(cid:5)isg-goodin(cid:3).Thespace ofsuch
measures is denotedby Mg.(cid:3);(cid:4)/. Similarly, if (1) has a solution for f D 0 and a
measurehD(cid:6) 2M.@(cid:3)/,wesaythat(cid:6)isg-goodon@(cid:3).Thespaceofsuchmeasures
is denoted by Mg.@(cid:3)/. If Mg.(cid:3);(cid:4)/ D M.(cid:3);(cid:4)/ we say that the nonlinearity g is
subcriticalintheinterior.Similarly,ifMg.@(cid:3)/DM.@(cid:3)/wesaythatgissubcritical
relativetotheboundary.
InSection2.2wepresentbasicresultsonboundaryvalueproblemswithmeasures.
Forinstance,assumingthat(cid:5)and(cid:6)areg-good,weshowthat(1)withf D(cid:5),hD(cid:6)
hasauniqueweaksolutionuandderiveestimatesforkuk andkgıuk in
L1.(cid:2)/ L1.(cid:2);(cid:3)/
termsofthenormsof(cid:5)and(cid:6) intheirrespectivespaces.Inparticularwefindthat,if
asolutionexistsitisunique.
Animportanttoolinourstudyisanextensionof themethodofsub-andsuperso-
lutionstothecase ofweak solutionsandageneralclassofnonlinearities.Thistoois
presentedinSection2.2
InSection2.3we presentasufficientconditionfor interiorandboundarysubcriti-
cality.It isshownthatthisconditionalsoimpliesstabilitywith respect toweak con-
vergenceofdata.Further,inSection2.4,wediscussthestructureofthespaceofgood
measureswhenthe nonlinearityg issupercritical intheinterior (resp. onthebound-
ary),i.e.,Mg.(cid:3);(cid:4)/ M.(cid:3);(cid:4)/(resp.Mg.@(cid:3)/ M.@(cid:3)/).
Chapter3isdevotedtoastudyoftheboundarytraceproblemforpositivesolutions
oftheequation
(cid:2)(cid:2)uCg.x,u/D0, (4)
withg asin(1),andrelatedboundaryvalueproblems.Thebasicmodelforourstudy
istheboundarytracetheoryforpositiveharmonicfunctionsduetoHerglotz.
ByHerglotz’s theoremanypositiveharmonicfunctioninaboundedLipschitzdo-
mainadmitsaboundarytraceexpressedbyaboundedmeasureandtheharmonicfunc-
tionisuniquelydeterminedbythistraceviaanintegralrepresentation.
Preface vii
Thenotionofaboundarytraceofafunctionuin(cid:3)dependsontheregularityprop-
ertiesofthefunction.Forinstance,ifu 2C.(cid:3)N/thenithasaboundarytraceinC.@(cid:3)/,
namely,ub .IfubelongstoaSobolevspaceW1,p.(cid:3)/forsomep >1thenithasa
@(cid:2)
boundarytraceinLp.@(cid:3)/(andeveninamoreregularspace,namely,W1(cid:2)p1,p.@(cid:3)/).
Themeasureboundarytraceofapositiveharmonicfunctionisdefinedasfollows:let
¹(cid:3) ºbeanincreasingsequenceofdomainsconvergingto(cid:3);undersomerestrictions
n
onthissequenceitcanbeshownthatthesequenceofmeasures¹ub dSºconverges
weaklyinM.(cid:3)N/(=thespaceoffiniteBorelmeasuresin(cid:3)N)toame@a(cid:2)snure(cid:6) 2M.@(cid:3)/
thatisindependentof¹(cid:3) º.Thislimitingmeasureisthemeasureboundarytraceofu.
n
If(cid:3)isofclassC2theharmonicfunctionucanberecoveredfromitsmeasurebound-
arytraceviathePoissonintegral.IfthedomainismerelyLipschitz,thePoissonkernel
mustbereplacedbytheMartinkernel.(FormoredetailsseeSection1.3.)
As a first step in our study of the trace problem for positive solutions of (4) we
considermoderatesolutions.Apositivesolutionof(4) ismoderateifitisdominated
byaharmonicfunction.ThefollowingresultisaconsequenceoftheHerglotztheorem.
A positive solutionu is moderateif andonly if g ıu 2 L1.(cid:3);(cid:4)/. Every positive
moderatesolutionpossessesaboundarytracerepresentedbyaboundedmeasure.
Sofarthetrace problemforpositivesolutionsofthenonlinearequationappearsto
besimilartothetraceproblemforpositiveharmonicfunctions.However,beyondthis
similarity, the nonlinear problempresents two essentially new aspects. The first is a
fact already mentionedbefore: ingeneral, there existpositivefinite measures on@(cid:3)
thatarenotboundarytracesofanysolutionof(4).Thesecond:theequationmayhave
positivesolutionsthatdonothaveaboundarytraceinM.@(cid:3)/.
Bothaspectsarepresentinthebasicexamples(3).Inthecaseofpowernonlineari-
tiesg.t/ D jtjqsignt,if q (cid:3) .N C1/=.N (cid:2)1/andN (cid:3) 2thereisnosolutionwith
boundarytracegivenbyaDiracmeasure.Infactinthiscasethereisnosolutionwith
anisolatedsingularity.Inotherwords,isolatedpointsingularitiesareremovable.(For
detailsseeSubsection3.4.3and4.2.1.)
ThesecondaspectoccurswhenevergsatisfiestheKeller–Ossermanconditiondis-
cussedbelow.Thisconditionissatisfiedbypowernonlinearitiesforeveryq >1and
bytheexponentialnonlinearity.
J.B.Keller [60]andR.Osserman[96]providedasharpconditiononthegrowthof
gatinfinitywhichguaranteesthatthesetofsolutionsof(4)isuniformlyboundedfrom
aboveincompactsubsetsof(cid:3).Qualitativelytheconditionmeansthatthesuperlinear-
ityofgatinfinityissufficientlystrong.Assumingthatthisconditionholdsuniformly
withrespect tox 2 (cid:3),theyderivedana prioriestimate for solutionsof(4) interms
of(cid:4).x/ D dist.x,@(cid:3)/.Thisestimateimpliesthatequation(4),inboundeddomains,
possessesa maximal solution.If, inaddition,(cid:3) satisfies theclassical Wiener condi-
tionthenthemaximalsolutionblowsupeverywhereontheboundary.(Ifg.x,0/D 0
theboundednessassumptiononthedomainisnotneeded.)A solutionthatblowsup
everywhere onthe boundaryiscalled a largesolution.Evidently,large solutionsdo
notpossesaboundarytraceinM.@(cid:3)/.
viii Preface
In Section 3.1 we show that every positive solution has a boundary trace that is
givenbyanouterregular Borelmeasure; howeverthismeasureneednotbefinite.If
thesolutionismoderatethisreducestotheboundarytracepreviouslymentioned.The
boundarytrace (cid:5)N ofa positivesolutionuhasa singularsetF (possiblyempty)such
that (cid:5)N is infinite on F while (cid:5)N is a Radon measure on @(cid:3)n F. The singular set is
closed.Apointy 2 @(cid:3)issingular(relativetou)ify 2F andregularotherwise.The
singularandregularboundarypointsaredeterminedbyalocalintegralcondition.
A boundary trace (cid:5)N can also be represented by a couple .F,(cid:5)/ where F is the
singularsetofthetrace and(cid:5)isaRadonmeasureon@(cid:3)nF.Thesetofallpositive
measuresthatcanberepresentedinthismannerisdenotedbyB .Asolutionwhose
reg
boundarytraceisoftheform.F,0/iscalledapurelysingularsolution.
Assumimg that the Keller–Osserman condition holds uniformly in (cid:3), for every
compactsetF (cid:5)@(cid:3)thereexistsasolutionU thatismaximalinthesetofsolutions
F
vanishingon@(cid:3)nF (seeSection3.2).U iscalledthemaximalsolutionrelativeto
F
F.Inthesubcriticalcase,theboundarytraceofU is.F,0/.Inthesupercriticalcase,
F
the singular set of U – denotedby k .F/ – may be smaller than F. The maximal
F g
solutionsU playacrucialroleinthestudyoftheboundaryvalueproblem
F
(cid:2)(cid:2)uCg.x,u/D 0 in(cid:3)
(5)
uD (cid:5)N on@(cid:3)
wheng 2 G and(cid:5)N 2 B .
0 reg
InSection3.3wepresentageneralresultprovidingnecessaryandsufficientcondi-
tionsforexistenceanduniquenessofsolutionsof(5)assumingthatgsatisfiesthelocal
Keller–Ossermanconditionandtheglobalbarrierconditionandthat,foreveryx 2 (cid:3),
g.x,(cid:4)/isconvex.(Seedefinitions3.1.9and3.1.10.)Theseconditionsaresufficientfor
theexistenceofthemaximalsolutionU .
F
InSection3.4westudyproblem(5)whengisgivenby
g.x,t/ D(cid:4).x/ˇjtjq(cid:2)1t, q >1, ˇ >(cid:2)2. (6)
Assumingthat(cid:3)isasmoothdomainweshow:(i)Ag-barrierexistsateveryboundary
pointandtheglobalbarrier conditionholdsand(ii)g issubcriticalifandonlyif
1<q <q .˛/:D.N CˇC1/=.N (cid:2)1/.
c
Next we apply the result of Section 3.3to problem (5) with g as aboveassuming
thatqisinthesubcriticalrange.Weshowthat,undertheseassumptions:
Problem(5)possessesauniquesolutionforevery(cid:5)N 2 B .
reg
Therefollowsadescriptionofthemainstepsintheproofofthisresult:
I. Foreveryy 2@(cid:3)thereisauniquesolutionwithboundarytrace.¹yº,0/denoted
byu1,y.
Preface ix
II. IfuisasolutionwithsingularboundarysetF thenforeveryy 2F,
u(cid:3)u1,y.
Usingthesetworesultsweshowthat:
III. ForeverycompactF (cid:5)@(cid:3),themaximalsolutionU istheuniquesolutionwith
F
trace.F,0/.
Theproofiscompletedbyestablishingthefollowing:
IV. If,foreverycompactsetF (cid:5)@(cid:3),(5)hasauniquesolutionwithboundarytrace
.F,0/thentheboundaryvalueproblemhasauniquesolutionforeverymeasure
(cid:5)N 2 B .
reg
Twoparticularcasesoftheboundaryvalueproblem(5)havereceivedspecialattention
intheliterature.
Thefirstisthecaseoflargesolutionsalreadymentionedabove.Inthelanguageof
boundarytraces, thesingularboundarysetofalargesolutionisthewholeboundary.
In the case of Lipschitz domains, the global Keller–Osserman conditionimplies the
existence of a large solution.However, in moregeneral domains,the maximal solu-
tionmaynotblowupeverywhere ontheboundary.Therefore, insucha casea large
solutiondoesnotexist.
Thequestionofexistenceanduniquenessofalargesolutionundervariousassump-
tionsongand(cid:3)hasbeenasubjectofintensestudy.Inadditiontoitsintrinsicinterest,
thistopicisusefulindelineatingthelimitationsthatarenaturallyimposedonthegoals
ofourstudyofgeneralboundaryvalueproblems.
ThesubjectoflargesolutionsisdiscussedindetailinChapter5.
Thesecondcasetoreceivespecialattentionisthatofsolutionswithisolatedsingu-
larities. Ifthenonlinearityissubcritical then,foreveryy 2 @(cid:3)there existmoderate
solutionswithisolatedsingularityaty.If,inaddition,ag-barrierexistsatythenthere
existnon-moderatesolutionswithanisolatedsingularityaty.Suchasolutioniscalled
a‘verysingularsolution’.Alternativelywesaythatthesolutionhasa‘strongisolated
singularity’aty.
Assumethatg issubcriticalandthatag-barrier existsaty 2 @(cid:3).Letu denote
k,y
thesolutionwithboundarytracekı .Fork >0thissolutionisdominatedbykP.(cid:4),y/
y
(where P denotes thePoissonkernel); therefore it is a moderate solution.However,
theexistenceofabarrieraty impliesthat
u1,y D lim uk,y (7)
k!1
isasolutionoftheequationwhichvanisheson@(cid:3)n¹yº.Evidentlythissolutionhasa
strongsingularityaty.Theanalysisofthesetofsolutionswithstrongisolatedsingular-
itiesplaysanimportantroleinthestudyofboundaryvalueproblemsinthesubcritical
case. A questionofspecial interest isthe uniquenessof the verysingular solutionat
