Table Of ContentUniversitext
Thomas Alazard
Claude Zuily
Tools and
Problems
in Partial
Differential
Equations
Universitext
Universitext
SeriesEditors
SheldonAxler
SanFranciscoStateUniversity
CarlesCasacuberta
UniversitatdeBarcelona
JohnGreenlees
UniversityofWarwick
AngusMacIntyre
QueenMaryUniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
ÉcolePolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski
CaseWesternReserveUniversity
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Thomas Alazard (cid:129) Claude Zuily
Tools and Problems in Partial
Differential Equations
ThomasAlazard ClaudeZuily
ÉcoleNormaleSupérieureParis-Saclay InstitutdeMathématiqued’Orsay
UniversitéParis-Saclay UniversitéParis-Saclay
GifsurYvette,France Orsay,France
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-3-030-50283-6 ISBN978-3-030-50284-3 (eBook)
https://doi.org/10.1007/978-3-030-50284-3
MathematicsSubjectClassification:35,76
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To ourfamilies.
Introduction
Theaimofthisbookistopresent,through65fullysolvedlongproblems,various
aspects of the currenttheory of partialdifferentialequations(PDE). It is intended
forgraduatestudentswhowouldlike,throughpractice,totesttheirunderstanding
ofthetheory.
Thebookismadeoftwoparts.Thefirstpart(Chaps.1–8)containssometheory
(intheoddchapters)andthestatementsoftheproblems(intheevenchapters).The
secondpartcontainsthesolutions(Chap.9)andanappendix(Chap.10),wherewe
recallsomebasicsoftheclassicalanalysis.
Even though the main purpose of this book is to present these problems and
their solutions, for the reader’s convenience, we have recalled some of the main
theoreticalresults concerningeach topic. This is whyeach chapter of problemsis
precededby a shortintroductionrecallingwithoutproofthe basic facts. However,
somecommentsindicatewhereonemayfindthedetailsoftheproofs.Thismakes
thebookessentiallyself-containedforthereader.
Since the theoryofpartialdifferentialequationsis a verywide subject, it is by
nomeansrealistictohopetodescribeallthetopicsina singlevolume.Therefore,
choiceshavetobemadeandwehavechosentofocusonafewofthem.
Letusnowdescribemorepreciselythecontentsofthisbook.
Thefirstchapterisintroductory.Someoftheessentialtoolsinfunctionalanalysis
which are commonly used in PDE are recalled. This includes the main theorems
concerning Fréchet, Banach, or Hilbert spaces. We have also recalled the main
notionsin distributionstheory,includingthe analysisofthe Fouriertransformand
thestationaryphaseformula.
The second chapter contains three problems on the subjects discussed in the
previousone.
The third chapter begins with the description of some of the main function
spacesusedinPDE,namelySobolev,Hölder,andZygmundspaces(includingtheir
Littlewood–Paley characterization). Other tools, such as interpolation theory and
paraproducts,arealsopresented.
vii
viii Introduction
Inthefourthchapter,through15problems,wediscussvariousapplicationssuch
asfunctionalinequalities(productandcompositionrulesinSobolevspaces,Hardy
inequality,etc.) andwe introduceotherfunctionspacessuch as spacesof analytic
functions,uniformlylocalSobolevspaces,weakLebesguespaces,spaceofbounded
meanoscillation,etc.
Thefifthchapterisconcernedwiththetheoryofmicrolocalanalysis.Wereview
the main notions about the pseudo-differential and paradifferential operators, the
wavefrontset,andthemicrolocaldefectmeasuresandwedescribesomeofthemain
results,suchascontinuity,Gardinginequalities,andpropagationofsingularities.
In the sixth chapter, through 18 problems, we give some applications of these
notions, in particular we explore the notions of symbols, hypoelliptic operators,
smoothingeffect,andCarlemaninequalities.
Theseventhchapterreviewsthemainclassicalpartialdifferentialequationsthat
is the Laplace, wave, Schrödinger,heat, Burgers, Euler, Navier–Stokes equations,
andtheirmainpropertiesarerecalled.
Thelastchapterofthefirstpartofthebookcontains29problemsontheabove
equations. For instance, several problems about spectral theory for the Laplace
equation, about linear and nonlinear wave and Schrödinger equations are stated.
Moreover, other equations such as Monge–Ampère, the mean-curvature, kinetic,
andBenjamin–Onoequationsarediscussed.
The secondpart of the book containsthe detailed solutionsto all the problems
and a chapter gathering several fundamental results concerning the basics of
classical analysis, such as Lebesgue integration, differential calculus, differential
equationsandholomorphicfunctions.
Acknowledgements
We would like to warmly thank the anonymousreferees for their excellent work,
whichhelpedustoimprovethepresentationofthisbook.
However,itshouldbeclearthattheauthorsaresolelyresponsibleofanymistake
remaining. A list of possible corrections will be available at http://talazard.perso.
math.cnrs.fr.
After this bookwas proposedto Springer,we had severalmail exchangeswith
Mr Rémi Lodh. We would like to warmly thank him for his suggestions, support,
andefficiency.
April2020 ThomasAlazard
ClaudeZuily
Contents
PartI ToolsandProblems
1 ElementsofFunctionalAnalysisandDistributions..................... 3
1.1 FréchetSpaces........................................................ 3
1.2 ElementsofFunctionalAnalysis.................................... 5
1.2.1 FixedPointTheorems...................................... 5
1.2.2 TheBanachIsomorphismTheorem ....................... 5
1.2.3 TheClosedGraphTheorem................................ 5
1.2.4 TheBanach–SteinhausTheorem........................... 6
1.2.5 TheBanach–AlaogluTheorem ............................ 6
1.2.6 TheAscoliTheorem........................................ 7
1.2.7 TheHahn–BanachTheorem ............................... 7
1.2.8 HilbertSpaces .............................................. 7
1.2.9 SpectralTheoryofSelf-AdjointCompactOperators..... 9
1.2.10 Lp Spaces,1≤p ≤+∞ ................................. 9
1.2.11 TheHölderandYoungInequalities........................ 10
1.2.12 ApproximationoftheIdentity ............................. 11
1.3 ElementsofDistributionTheory .................................... 12
1.3.1 Distributions ................................................ 12
1.3.2 TemperedDistributions..................................... 13
1.3.3 TheFourierTransform ..................................... 15
1.3.4 TheStationaryPhaseMethod.............................. 16
2 StatementsoftheProblemsofChap.1................................... 19
3 FunctionalSpaces .......................................................... 25
3.1 SobolevSpaces....................................................... 25
3.1.1 SobolevSpacesonRd,d ≥1.............................. 25
3.1.2 LocalSobolevSpacesHs (Rd) .......................... 28
loc
3.1.3 SobolevSpacesonanOpenSubsetofRd................. 29
3.1.4 SobolevSpacesontheTorus............................... 31
ix
x Contents
3.2 TheHölderSpaces ................................................... 31
3.2.1 HölderSpacesofIntegerOrder............................ 31
3.2.2 HölderSpacesofFractionalOrder......................... 32
3.3 CharacterizationofSobolevandHölderSpaces
inDyadicRings ...................................................... 33
3.3.1 CharacterizationofSobolevSpaces ....................... 34
3.3.2 CharacterizationofHölderSpaces......................... 34
3.3.3 TheZygmundSpaces....................................... 35
3.4 Paraproducts.......................................................... 35
3.5 SomeWordsonInterpolation........................................ 37
3.6 TheHardy–Littlewood–SobolevInequality......................... 39
4 StatementsoftheProblemsofChap.3................................... 41
5 MicrolocalAnalysis......................................................... 61
5.1 SymbolClasses....................................................... 61
5.1.1 DefinitionandFirstProperties............................. 61
5.1.2 Examples.................................................... 62
5.1.3 ClassicalSymbols .......................................... 62
5.2 Pseudo-DifferentialOperators....................................... 62
5.2.1 DefinitionandFirstProperties............................. 62
5.2.2 Kernelofa(cid:2)DO ........................................... 63
5.2.3 Imageofa(cid:2)DObyaDiffeomorphism ................... 63
5.2.4 SymbolicCalculus.......................................... 64
5.2.5 Actionofthe(cid:2)DOonSobolevSpaces ................... 65
5.2.6 GardingInequalities........................................ 65
5.3 InvertibilityofEllipticSymbols..................................... 66
5.4 WaveFrontSetofaDistribution .................................... 66
5.4.1 DefinitionandFirstProperties............................. 66
5.4.2 WaveFrontSetand(cid:2)DO.................................. 67
5.4.3 ThePropagationofSingularitiesTheorem................ 67
5.5 ParadifferentialCalculus............................................. 68
5.5.1 SymbolsClasses............................................ 68
5.5.2 ParadifferentialOperators.................................. 69
5.5.3 TheSymbolicCalculus..................................... 70
5.5.4 LinkwiththeParaproducts................................. 71
5.6 MicrolocalDefectMeasures......................................... 71
6 StatementsoftheProblemsofChap.5................................... 75
7 TheClassicalEquations ................................................... 101
7.1 EquationswithAnalyticCoefficients ............................... 101
7.1.1 TheCauchy–KovalevskiTheorem......................... 102
7.1.2 TheHolmgrenUniquenessTheorem...................... 103
