Universitext 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 Universitext is a series of textbooksthat presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,intoverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Moreinformationaboutthisseriesathttp://www.springer.com/series/223 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2020 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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