Lecture Notes in Mathematics 2218 Darya Apushkinskaya Free Boundary Problems Regularity Properties Near the Fixed Boundary Lecture Notes in Mathematics 2218 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Princeton AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Darya Apushkinskaya Free Boundary Problems Regularity Properties Near the Fixed Boundary 123 DaryaApushkinskaya DepartmentofMathematics SaarlandUniversity Saarbru¨cken Germany ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-97078-3 ISBN978-3-319-97079-0 (eBook) https://doi.org/10.1007/978-3-319-97079-0 LibraryofCongressControlNumber:2018952878 MathematicsSubjectClassification(2010):Primary:35R35;Secondary:35R37,35B65,35K10,35J15 ©SpringerNatureSwitzerlandAG2018 TranslationfromtheEnglishlanguageedition:FreeBoundaryProblems:RegularityPropertiesNearthe FixedBoundarybyDaryaApushkinskaya,Copyright©SpringerInternational PublishingAG,partof SpringerNature2018.AllRightsReserved. Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Free boundary problems (FBPs) belong to the most striking component of the moderntheoryofpartialdifferentialequations(PDEs).TheexpressionFBPrefersto aprobleminwhichoneorseveralvariablesaregovernedindifferentsubdomainsof thespace,orspace-time,bythedifferentstatelaws.Thesesubdomainsareapriori unknown and have to be determined as a part of the problem. The boundariesof theseunknownsubdomainsarecalledthefreeboundaries. FBPs are the typical example of nonlinear problems where singularities arise. Therefore,aparticulardirectioninFBPshasbeentostudytheregularityproperties of solutions and those of the free boundaries. Such questions are important for experiments and numerics and are usually considered extremely hard. Since the free boundary is not known a priori, the classical techniques of elliptic/parabolic PDEsdonotapply.Inthelasttwo decades,manynewapproaches,combiningthe ideasfromPDEswithonesfromgeometricmeasuretheory,calculusofvariations, harmonicanalysis,andsoon,havebeendevelopedandprovidedinterestingresults. Thisbooktreatssomepartsofthissubjectanditsrecentdevelopment. To be precise, this book is devoted to the so-called obstacle-type problems, a classofFBPswhichmaybecharacterizedbythefollowingproperty:gradientofa solution is continuousacross the interface. For several elliptic and parabolic one- and two-phase obstacle-type problems, the qualitative properties of solutions and freeboundariesnearthefixedboundaryofadomainarestudied.Itissupposedthat theDirichletdataareprescribedonthefixedboundary. Thebookisdividedintothreechaptersandthreeappendices.Figure1illustrates themaindependenciesamongthebook’scomponents. Chapter 1 is introductoryin nature.Itcontainsa narrativeintroductioninto the fieldofFBPsaswellasabriefobservationofotherpartsofthebookandtheoutlines ofthemainsteps. Chapters2and3formthecoreofthebook.Theycanbetreatedindependently. The largestpart of the text contains results on parabolic problems.It includesthe wholeChap.2andahalfofChap.3. In Chap.2, the complete study of regularity issues up to the fixed boundaryis carried out for the case of the no-sign parabolic obstacle-type problem with the v vi Preface Fig.1 Themain dependenciesamongchapters Chapter 1 Appendix A Chapter 2 Chapter 3 Appendix B Appendix C homogeneous Dirichlet boundary data. As the first step the optimal regularity of solutionsisestablished;afterthattheanalysisofglobalsolutionsisgiven.Basedon theseresults,thefinepropertiesofthefreeboundaryareobtained,whichboildown toprovingaparabolic-tangentialtouchbetweenthefreeandfixedboundaries.The latterinturncanbeusedtoshowC1 propertiesofthefreeboundary. InthecaseofthenonhomogeneousDirichletboundarydata,thesituationismuch morecomplicated,especiallyforthetwo-phaseproblems.Eventoprovetheoptimal regularityofsolutionsisnoteasy;itrequiresaspecialcontrolfordependenceofall the estimateson the distanceto the fixedboundary.Thesequestionsarediscussed inChap.3.Someoftheresultsobtainedtherearestrongerthanonesknownforthe classicalobstacleproblem. In Appendix A, one can find all necessary information about various mono- tonicityformulaswhicharethemostimportanttechnicaltoolsinstudyingthefree boundary problems. In Appendix B, we recall and explain several general facts. Most of these auxiliary results are known, but probably not well known in the contextusedinthisbook.Forthereader’sconvenience,we collectinAppendixC some facts concerning various problems with free boundaries. These facts are involvedessentiallyinourarguments.Alltheseappendicesareverymuchinregular useinChaps.2and3. The intended audience of this book includes graduate students and young researchers entering this field of mathematics. The reader is assumed to be familiar with classical calculus and the standard elliptic/parabolic theory (maxi- mum/comparison principle, interior/boundary estimates, compactness arguments, etc.). This text is based on the revised version of my habilitation thesis at Saarland University. Parts of the book have been used as material for graduate courses on FBPs that I taughtat Saarland University(2010,2011, and 2017) and at Peoples’ FriendshipUniversityofRussiainMoscow(2016). I would like to mention just a few names standing for the long list of persons who contributed to this book in one way or the other. First and foremost, I am Preface vii deeplyindebptedtoNinaUraltseva.SheledmetotheDiplomaandPhD.Wehave been collaborating for more than 20 years. I am very grateful to her for support, numerousdiscussionsandvaluableadvice.MyspecialthanksgotoMartinFuchs. Without his kind supportand sometimes also his pressure, this book would never havebeencompleted.Furthermore,IwishtothankMichaelBildhauerwhocarefully read the manuscript and gave me useful suggestions. I would like to express my sincerethankstomycoauthorsHenrikShahgholianandNorayrMatevosyanforthe contributionstheyhavemadetoourjointpublications.Finally,Iamverythankful to myfamilyfor theireffortsto notdisturbme too muchand sharingwith me the pressureassociatedwithaprojectlikethis. I thank the Mathematical Sciences Research Institute (MSRI), Berkeley, USA, for hosting a program on Free Boundary Problems, Theory and Applications in Spring2011,whereIwasinresidenceandwroteportionsofthisbook. It remains only to note that this work was partially supported by the Russian Foundation of Basic Research (RFBR) through the grant 17-01-00099 and by the German-Russian InterdisciplinaryScience Center (G-RISC) throughthe grant M-2016b-3. Saarbrücken,Germany DaryaApushkinskaya July2018 Contents 1 Introduction .................................................................. 1 1.1 No-SignParabolicObstacle-TypeProblems.......................... 4 1.1.1 Motivation...................................................... 4 1.1.2 MainResultsforNo-SignParabolicObstacle-Type Problems........................................................ 7 1.1.3 HistoricalReview .............................................. 8 1.1.4 MainStrategy .................................................. 9 1.2 BoundaryEstimatesforSolutionsofFreeBoundaryProblems...... 12 1.2.1 EstimatesforSolutionstotheEllipticObstacleProblem.... 12 1.2.2 EstimatesforSolutionstotheTwo-PhaseElliptic Problem......................................................... 15 1.2.3 EstimatesforSolutionstotheTwo-PhaseParabolic Problem......................................................... 20 1.2.4 EstimatesNeartheInitialStateforSolutionstothe Two-PhaseParabolicProblem................................. 22 1.3 Appendices ............................................................. 25 1.3.1 AppendixA:MonotonicityFormulas......................... 25 1.3.2 AppendixB:AuxiliaryResults................................ 25 1.3.3 AppendixC:AdditionalFacts................................. 26 1.4 Notes.................................................................... 26 1.5 OpenProblems ......................................................... 26 2 No-SignParabolicObstacle-TypeProblems.............................. 29 2.1 StatementoftheProblemandMainResults .......................... 29 2.2 OptimalRegularityofSolutions....................................... 33 2.3 UsefulPropertiesofSolutions......................................... 44 2.3.1 Nondegeneracy................................................. 44 2.3.2 MeasureofΓ(u)............................................... 47 2.3.3 Convergence.................................................... 48 2.3.4 Blow-UpandBlow-Down..................................... 49 2.3.5 BalancedEnergy ............................................... 51 ix x Contents 2.4 ClassificationoftheNonnegativeGlobalSolutions .................. 53 2.5 Geometric Classification of the Global Solutions with NoSignRestrictions ................................................... 57 2.6 CharacterizationoftheFreeBoundaryPointsNearΠ ............... 60 2.7 RegularityPropertiesofSolutions..................................... 63 2.8 RegularityPropertiesoftheFreeBoundary........................... 70 3 BoundaryEstimatesforSolutionsofFreeBoundaryProblems........ 73 3.1 One-SidedEstimates up to the Boundaryfor Solutions totheEllipticObstacleProblem....................................... 73 3.1.1 StatementoftheProblemandMainResults.................. 73 3.1.2 EstimatesforMixedDerivativesontheBoundary ........... 75 3.1.3 EstimatesforPureSecondDerivatives........................ 80 3.2 BoundaryEstimatesforSolutionstotheTwo-PhaseElliptic Problem................................................................. 85 3.2.1 StatementoftheProblemandMainResult................... 85 3.2.2 EstimatesoftheTangentialGradientNeartheBoundary.... 86 3.2.3 BoundaryEstimatesoftheSecondDerivatives............... 89 3.3 EstimatesNeartheGivenBoundaryoftheSecond-Order DerivativesforSolutionstotheTwo-PhaseParabolicProblem...... 92 3.3.1 StatementoftheProblemandMainResult................... 92 3.3.2 Lipschitz Estimate of the Normal Derivative attheBoundaryPoints......................................... 93 3.3.3 BoundaryEstimatesoftheSecondDerivatives............... 95 3.4 UniformEstimates Near the Initial State for Solutions totheTwo-PhaseParabolicProblem.................................. 98 3.4.1 StatementoftheProblemandMainResult................... 98 3.4.2 EstimateoftheTimeDerivative............................... 100 3.4.3 EstimatesoftheSecondDerivatives .......................... 102 A MonotonicityFormulas ..................................................... 107 A.1 C-monotonicityFormula............................................... 107 A.2 ACF-monotonicityFormula ........................................... 113 A.3 W-monotonicityFormula .............................................. 115 B AuxiliaryResults............................................................. 123 C AdditionalFacts.............................................................. 131 References......................................................................... 137 Index............................................................................... 143