Lecture Notes of the Unione Matematica Italiana Pascal Cherrier Albert Milani Evolution Equations of von Karman Type Lecture Notes of 17 the Unione Matematica Italiana Moreinformationaboutthisseriesathttp://www.springer.com/series/7172 EditorialBoard CiroCiliberto(EditorinChief) CH-8057Zuerich,Switzerland DipartimentodiMatematica e-mail:[email protected] Universita’diRomaTorVergata FrancoFlandoli ViadellaRicercaScientifica DipartimentodiMatematicaApplicata 00133Roma(Italia) UniversitàdiPisa e-mail:[email protected] ViaBuonarroti1c SusannaTerracini(Co-editorinChief) 56127Pisa,Italy UniversitàdegliStudidiTorino e-mail:fl[email protected] DipartimentodiMatematica“GiuseppePeano” AngusMcintyre ViaCarloAlberto10 QueenMaryUniversityofLondon 10123Torino,Italy SchoolofMathematicalSciences e-mail:[email protected] MileEndRoad AdolfoBallester-Bollinches LondonE14NS Departmentd’Àlgebra UnitedKingdom FacultatdeMatemàtiques e-mail:[email protected] UniversitatdeValència GiuseppeMingione Dr.Moliner,50 DipartimentodiMatematicaeInformatica 46100Burjassot(València) UniversitàdegliStudidiParma Spain ParcoAreadelleScienze,53/a(Campus) e-mail:[email protected] 43124Parma,Italy AnnalisaBuffa e-mail:[email protected] IMATI–C.N.R.Pavia MarioPulvirenti ViaFerrata1 DipartimentodiMatematica, 27100Pavia,Italy UniversitàdiRoma“LaSapienza” e-mail:[email protected] P.leA.Moro2 LuciaCaporaso 00185Roma,Italy DipartimentodiMatematica e-mail:[email protected] UniversitàRomaTre FulvioRicci LargoSanLeonardoMurialdo ScuolaNormaleSuperiorediPisa I-00146Roma,Italy PiazzadeiCavalieri7 e-mail:[email protected] 56126Pisa,Italy FabrizioCatanese e-mail:[email protected] MathematischesInstitut ValentinoTosatti Universitätstraße30 NorthwesternUniversity 95447Bayreuth,Germany DepartmentofMathematics e-mail:[email protected] 2033SheridanRoad CorradoDeConcini Evanston,IL60208 DipartimentodiMatematica USA UniversitàdiRoma“LaSapienza” e-mail:[email protected] PiazzaleAldoMoro5 CorinnaUlcigrai 00185Roma,Italy ForschungsinstitutfürMathematik e-mail:[email protected] HGG44.1 CamilloDeLellis Rämistrasse101 InstitutfuerMathematik 8092Zürich,Switzerland UniversitaetZuerich e-mail:[email protected] Winterthurerstrasse190 TheEditorialPolicycanbefoundatthebackofthevolume. Pascal Cherrier (cid:129) Albert Milani Evolution Equations of von Karman Type 123 PascalCherrier AlbertMilani DépartmentdeMathématiques DepartmentofMathematics UniversitéPierreetMarieCurie UniversityofWisconsin Paris,France Milwaukee,WI,USA ISSN1862-9113 ISSN1862-9121 (electronic) LectureNotesoftheUnioneMatematicaItaliana ISBN978-3-319-20996-8 ISBN978-3-319-20997-5 (eBook) DOI10.1007/978-3-319-20997-5 LibraryofCongressControlNumber:2015952855 MathematicsSubjectClassification(2010):53D05,53D12,37J05,37J10,35F21,58E05,53Z05 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 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,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) To our wives, Annickand Claudia TentasseIuvat Preface In these notes, we consider two kinds of nonlinear evolution problems of von Karman type on R2m, m (cid:2) 2. Each of these problems consists of a system that resultsfromthecouplingoftwohighlynonlinearpartialdifferentialequations,one hyperbolic or parabolic, and the other elliptic. These systems are called “of von Karman type” because of a formal analogy with the well-known equationsof the samenameinthetheoryofelasticityinR2. 1 TheClassicalEquations 1. To describe the classical hyperbolic von Karman system in R2, we introduce the nonlinear coupling of the second order derivatives of two sufficiently smooth functionsgDg.x;y/andhDh.x;y/,definedby (cid:2) (cid:3) g g Œg;h(cid:2)WDdet xx xy ; (1) h h yx yy andthenweset N.g;h/WDŒg;h(cid:2)CŒh;g(cid:2)Dg h Cg h (cid:3)2g h : (2) xx yy yy xx xy xy TheclassicalvonKarmanequationsinR2consistofthesystem u C(cid:3)2uDN.f;u/CN.';u/; (3) tt (cid:3)2f D(cid:3)N.u;u/; (4) where (cid:3) the usual Laplace operator in R2, and ' D '.t;x;y/ is a given external source. Equations (3) and (4) model the dynamics of the vertical oscillations (buckling) of an elastic two-dimensional thin plate, represented by a bounded vii viii Preface domain(cid:4) (cid:4) R2,duetobothinternalandexternalstresses.Moreprecisely,inthis modeltheunknownfunctionuDu.t;x;y/isameasureoftheverticaldisplacement of the plate; Eq.(4) formally defines a map u 7! f.u/, where f.u/ represents the so-calledAiry stress function,which is related to the internalelastic forcesacting on the plate, and depends on its deformationu; finally, ' represents the action of the external stress forces. Typically, Eqs.(3)C(4) are supplemented by the initial conditions u.0/Du0; ut.0/Du1; (5) whereu0andu1areagiveninitialconfigurationofthedisplacementanditsvelocity, andbyappropriateconstraintsonuattheboundaryof(cid:4). 2.Adetailedandprecisedescriptionofthemodelingissuesrelatedtotheclassical vonKarmanequations,andadiscussionoftheirphysicalmotivations,canbefound in, e.g.,Ciarlet andRabier [12], or in Ciarlet [10,11];in addition,we refer to the recent, exhaustive study by Chuesov and Lasiecka [9] of a large class of initial- boundaryvalueproblemsofvonKarmantypeondomainsofR2,withamultitude ofdifferentboundaryconditions,includingnonlinearones.Thestationarystate of theclassicalvonKarmanequations,describedbythenonlinearellipticsystem (cid:3)2u DN.f;u/CN.';u/; (6) (cid:3)2f D(cid:3)N.u;u/; (7) has been investigated by several authors; in particular, Berger [3], devised a remarkable variational method to establish the existence of suitably regular solu- tions to the stationary system (6)C(7) in a bounded domain of R2, subject to appropriate boundary conditions. Weak solutions of the corresponding system of evolution (3)C(4)C(5), again under appropriate boundary conditions, have been established,amongothers,byLions[21,Chap.1,Sect.4],andFavinietal.[15,16], andChuesovandLasiecka[9]. 2 TheGeneralized Equations 1. To introduce the generalization of the von Karman system (3)C(4) we wish to consider,we nowletm 2 N(cid:2)2, and,givenmC1 smoothfunctionsu1; ::: ;um; u definedonR2m,weset N.u1; ::: ;um/ WD ıji11(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)jimmrij11u1 (cid:5)(cid:5)(cid:5) rijmmum; (8) M.u/ WD N.u; ::: ;u/DmŠ(cid:5) .r2u/; (9) m 2 TheGeneralizedEquations ix wherewe adoptthe usualsummationconventionforrepeatedindices,anduse the following notations. For i1; ::: ;im, j1; ::: ;jm 2 f1; ::: 2mg, ıji11(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)jimm denotes the Kronecker tensor; for 1 (cid:6) i; j (cid:6) 2m, rj WD @@, and (cid:5) is the m-th elementary i i j m symmetricfunctionoftheeigenvalues(cid:6) D(cid:6) .@@u/,1(cid:6)k (cid:6)2m,oftheHessian k k i j matrixH.u/WDŒ@@u(cid:2),thatis, i j X (cid:5) .r2u/WD (cid:6) (cid:5)(cid:5)(cid:5) (cid:6) : (10) m k1 km 1(cid:4)k1<k2<:::<km(cid:4)2m Wealsointroducetheconvention (cid:4) (cid:5) N u.1k1/; ::: ;u.pkp/ WDN.„u1; :ƒ:‚: ;u…1; ::: „up; :ƒ:‚: ;u…p/; (11) k1factors kpfactors withk1C (cid:5)(cid:5)(cid:5) Ckp Dm,andset(cid:3)WD(cid:3)rjju. InLemma1.3.1ofChap.1,weshallshowthattheellipticequation (cid:3)mf D(cid:3)M.u/ (12) canbeuniquelysolved,in asuitablefunctionalframe,forf intermsofu,thereby defining a map u 7! f WD f.u/. Let T > 0. Given a source term ' defined on Œ0;T(cid:2) (cid:7) R2m, we consider the Cauchy problem, of hyperbolic type, in which we wishtodetermineafunctionuonŒ0;T(cid:2)(cid:7)R2m,satisfyingtheequation u C(cid:3)muDN.f.u/;u.m(cid:5)1//CN.'.m(cid:5)1/;u/; (13) tt andsubjecttotheinitialconditions(5), where,now,u0 andu1 aregivenfunctions definedonR2m.WerefertothisCauchyproblem,thatis,explicitly,to(13)C(12)C (5),as“problem(VKH)”. Problem(VKH)appearstobeanalogoustotheoriginalvonKarmansystem(3) and(4)onR2,butthisanalogyisonlyformal,inthefollowingsense.Letddenote the space dimension. In the linear part at the left side of Eqs.(3) and (4) of the original system, the order of the differential operator (cid:3)2 is twice the dimension of space (i.e., 4 D 2d, d D 2), and the nonlinear operators of Monge-Ampère typeattherightsideoftheequationsaredefinedintermsofthecompleteHessian of functions depending on u, f, and '. In contrast, at the left side of Eqs.(13) and(12)theorderofthedifferentialoperator(cid:3)mequalsthedimensionofspace(i.e., 2m D d), while the Monge-Ampèreoperatorsat the right side of these equations aredefinedintermsofelementarysymmetricfunctionsoforderm D d ofHessian 2 matricesoffunctionsdependingonu,f,and'.Toillustratethisdifferenceexplicitly, intheoriginalequation(4)thetermN.u;u/istwicethedeterminantoftheHessian