Divergence Operator and Related Inequalities

132 Pages·2017·1.618 MB·English

by  Gabriel Acosta, Ricardo G. Durán

SPRINGER BRIEFS IN MATHEMATICS
Gabriel Acosta
Ricardo G. Durán
Divergence
Operator
and Related
Inequalities
123
SpringerBriefs in Mathematics
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Gabriel Acosta • Ricardo G. Dura´n
Divergence Operator
and Related Inequalities
123
GabrielAcosta RicardoG.Dura´n
DepartmentofMathematicsandIMAS DepartmentofMathematicsandIMAS
UniversityofBuenosAiresandCONICET UniversityofBuenosAiresandCONICET
BuenosAires,Argentina BuenosAires,Argentina
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-1-4939-6983-8 ISBN978-1-4939-6985-2 (eBook)
DOI10.1007/978-1-4939-6985-2
LibraryofCongressControlNumber:2017934519
MathematicsSubjectClassification(2010):26D10,76D07,42B20,46E35,35Q30
©TheAuthor(s)2017
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In memoryof ourdear friendandcolleague,
Mar´ıaAmeliaMuschietti.
Preface
GivenadomainΩandafunction f withvanishingmeaninΩ,isitpossibletofind
azerotracefieldusuchthat
divu= f
and
(cid:2)u(cid:2)W1,p(Ω)≤C(cid:2)f(cid:2)Lp(Ω)?
Iftheanswerispositive,howirregularcanΩbe?Andifitisnegative,isitpossible
to enlarge the space of involved functionsin which u is sought in order to obtain
solutions?
Thisproblem,thatis,theexistenceofarightinverseofthedivergenceoperator,
also called div in this book, is strongly related to the Korn inequality and to the
p
Stokes equations among many other relevant problems and inequalities. The pur-
poseofthisbookistopresentinaunifiedandcoherentwayanoverviewofmany
ofthesecentraltopics.
Ourapproachfollowssomehowourownlineofresearchinthesematterscollect-
ingmostofourproductionrelatedto thefield. Notwithstandingthat,manyproofs
havebeenimprovedorshortenedandthematerialorganizedina coherentfashion
thathopefullyhelpsthereadertogetsmoothlyintothesubject.Moreover,ateach
stage,wetriedtosurveyasmanycontributionsaswecouldcoveringbothclassical
papersandrecentdevelopments.Naturally,lengthandtimerestrictionsnecessarily
imposedalimit.Wetriedtofocusonspecificaspectsandthereforethereadermay
-andsurelywill- findomissionsthatgenuinelyshoulddeservetohavea placein
thisbook.Ourapologies,inadvance,forthem.
The book is organized as follows: several equivalences of the problem of ex-
istenceofarightinverseofthedivergenceareexploredinChapter1.Chapter2is
devotedtopresenttheconstructionofsolutionsfordiv elaboratedbyBogovskiifor
p
star-shapeddomainsandtoshowhowthatresultcanbegeneralizedtomoreintri-
catedomains,specificallytoJohndomains.Besidesgeometricaltechnicalities,the
mainideasarethesameandthekeytoolisultimately,inbothcases,theCaldero´n-
Zygmundsingularintegralstheory.Thelastsectionsofthatchaptertreattherelation
ofthatproblemwithimprovedPoincare´inequalities.Chapter3,inturn,focuseson
vii
viii Preface
Korn’s inequalities beginning with several equivalences, for the so-called second
caseoftheinequality,inarbitrarydomains.Then,Korn’sinequalitiesonJohndo-
mainsarederivedbydifferentalternativeapproachesusing,forinstance,theresults
previously obtained for the divergence operator as well as improved Poincare´ in-
equalities. The latter approach turns out to be useful even for obtaining weighted
versions of these inequalities on Ho¨lder-αdomains and for domains with exter-
nalcusps.ThisisextensivelytreatedalongChapter4wherealsofamiliesofcoun-
terexamplesareexhibitedtoprovethatthepresentedresultsaresharp.Moreover,in
Chapter4moreirregularcasesaretreated.Inthelastpart,cuspsareallowedtohave
roughboundaries,besidesthesingularpointatthetip.WeightedKorn’sinequalities
arealsoderivedforthemusingelementarydecompositiontechniques.
Finally,abriefandeveninformalderivationofsomebasicequationsofcontin-
uum mechanics and a short proof delimiting what powers of the distance to a set
functionbelongtotheMuckenhoupt’sA classtogetherwithsomesupplementary
p
materialisgiveninthreeappendices.
We would like to end this preface with just a few more words. We devoted, in
anonsystematicway,morethantwelveyearstotheissuestreatedinthisbook.We
didit, asit usuallyhappens,withouta master planin mindandfollowingmostof
thetimepersonaltastesandinterests.Lookingbackwenoticedthataveryrichnet
ofconnectionswassimplylostinthevastliteraturedealingwiththesetopics.Anet,
maybenotignoredbytheexpert,wewishedtoexposeclearlytoalargeraudience
alongthefollowingpages.
Last but not least, we want to thank our coauthors throughout all these years.
Without them, we would have never reached this point. When we were finishing
thisbookwereceivedthesadnewsthatourfriendMariaAmeliaMuschiettipassed
away.Animportantpartofthecontentspresentedherecorrespondstoworkdonein
collaborationwithher.Thisbookisdedicatedtohermemory.
BuenosAires,Argentina GabrielAcosta
December,2016 RicardoG.Dura´n
Contents
1 Introduction................................................... 1
2 DivergenceOperator ........................................... 13
2.1 SolutionsoftheDivergenceonStar-ShapedDomains ............ 14
2.2 SolutionsoftheDivergenceonJohnDomains................... 23
2.3 ImprovedPoincare´InequalityandEquivalences................. 38
2.4 APartialConverseResult.................................... 43
2.5 CommentsandReferences................................... 44
3 Korn’sInequalities ............................................. 47
3.1 ClassicalVersionsofKorn’sInequalities ....................... 48
3.2 TheKornInequalityonJohnDomains......................... 55
4 SingularDomains .............................................. 61
4.1 WeightedKornandPoincare´InequalitiesonHo¨lderαDomains ... 62
4.2 SolutionsoftheDivergenceinDomainswithExternalCusps...... 66
4.3 WeightedKornInequalitiesonExternalCusps .................. 71
4.4 SomeSimpleCounterexamples............................... 73
4.5 Korn anddiv forLinkedDomains........................... 79
p p
4.6 FromLinkedDomainstoMoreGeneralExternalCusps .......... 92
4.7 CommentsandReferences................................... 95
A BasicEquationsofContinuumMechanics ........................ 99
A.0.1 LinearizedElasticity .................................102
A.0.2 TheStokesEquations.................................104
ix
x Contents
B PowersofthedistancetotheboundaryasA weights...............107
p
C AnAuxiliaryResultonHo¨lderαDomains ........................109
References.........................................................115
Index .............................................................123
Notation and Preliminaries
WithΩ⊂Rn wedenoteageneraldomain(i.e.,anopenandconnectedset)which
is always assumed to be bounded.The diameter of Ωis denotedwith δ(Ω). The
standardLebesgueand Sobolevspaces in Ωare denotedrespectivelywith Lp(Ω)
andWk,p(Ω).Accordingly,wedefinefor1≤p<∞,
(cid:2)(cid:3) (cid:4)1/p
(cid:2)u(cid:2)Lp(Ω)= |u|pdx
Ω
andrecursively
(cid:2)u(cid:2)Wk,p(Ω)=(cid:2)u(cid:2)Wk−1,p(Ω)+ ∑ (cid:2)Dβu(cid:2)Lp(Ω),
|β|=k
whereβ=(β,···,β)withβ ∈N ,|β|=∑n β andDβ:= ∂|β| .Inthis
1 n i 0 i=1 i ∂xβ11xβ22···xβnn
context,Sobolevseminormsaregivenby
|u|Wk,p(Ω)= ∑ (cid:2)Dβu(cid:2)Lp(Ω).
|β|=k
Sobolevnormsandseminormsforp=∞aredefinedinthesamewayadoptingusual
modifications. We also write Hk(Ω):=Wk,2(Ω) and use Lp(Ω)n and W1,p(Ω)n
(resp.Lp(Ω)n×nandW1,p(Ω)n×n)forthevectorial(resp.tensorial)versionwiththe
naturalnormadaptation.Inordertohelpthereadability,elementsbelongingtothese
spacesarewritteninbold.Accordingly,wewriteinacompactform(cid:2)Du(cid:2)Lp(Ω)n×n
with Duthe differentialmatrixofu, usingsometimes(cid:2)∇u(cid:2)Lp(Ω)n forscalarfunc-
tions.Withinthiscontext,weuseu forthecomponentsofavectorialvariableu.Ac-
i
cordingly,wedenotethedivergencewithdivu=∑n ∂ui reservingthesymbolDiv
i=1 ∂xi
forthedivergenceofamatrixfieldM(x).ForeachelementinM(x)∈W1,p(Ω)n×n,
DivM∈Lp(Ω)n and is defined as DivM =∑n ∂Mij (i.e., the operator div is
i j=1 ∂xj
appliedalongeachrowofM(x)).
xi