Calculus of Variations for Differential Forms THÈSE NO 7060 (2016) PRÉSENTÉE LE 30 SEPTEMBRE 2016 À LA FACULTÉ DES SCIENCES DE BASE CHAIRE D'ANALYSE MATHÉMATIQUE ET APPLICATIONS PROGRAMME DOCTORAL EN MATHÉMATIQUES ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR Swarnendu SIL acceptée sur proposition du jury: Prof. J. Krieger, président du jury Prof. B. Dacorogna, directeur de thèse Prof. J. Kristensen, rapporteur Prof. P. Mironescu, rapporteur Prof. H.-M. Nguyen, rapporteur Suisse 2016 R´esum´e Dans cette th`ese, nous ´etudions le calcul des variations pour les formes diff´erentielles. Le premi`ere partie est d´edi´ee au d´eveloppement des outils des m´ethodes directes du calcul des variations pour r´esoudredes probl`emesde minimisation de fonctionnelles d’une ou plusieurs variables de la forme f (dω) , f (dω1, ..., dωw) , et f (dω, δω) . Ω Ω Ω Nous introduions les notions de convexit´eesappropri´ees `achaque cas, appel´ees polyconvexit´e ext., quasiconvexit´eext.,etun-convexit´eext.pour des fonctionnelles de la forme Ω f(dω), et la polyconvexit´eext. vectorielle,laquasiconvexit´eext. vectorielle,etlaun-convexit´e ext. vectorielle pour des fonctionnelles de la forme f(dω1,...,dω )ainsiquelapolyconvexit´e ext-int.,laqua- Ω m siconvexit´eext-int. et la un-convexit´eext-int. pour les fonctionnelles de la forme Ω f(dω, δω).. Nous ´etudions les liens et relations entre ces notions de convexit´e et leur homolgues du cas classique du calcul des variations, c’est-`a-dire,la polyconvexit´e, la quasiconvexit´eet la rang un convexit´e. Nous ´etudions ´egalement la semi-continuit´einf´erieure et la continuit´efaibledeces fonctionnelles sur des espaces appropri´eset nous nous occupons des probl`emesde coercivit´eet obtenons des th´eor`emes d’existence `adesprobl`emes de minimization de fonctionnelles d’une forme diff´erentielle. Dans la deuxi`eme partie, nous ´etudionsles probl`emesaux limites pour des op´erarteurs de type Maxwell lin´eaires, semi-lin´eraireset quasi-lin´eairespour des formes diff´erentielles. Nous ´etudions l’existence et ´etablissons la r´egularit´eint´erieure ainsi que des estimations pour la r´egularit´e L2 sur le bord pour l’op´erateur de MAxwell lin´eaire δ(A(x)dω)=f avec diff´erentes conditions au bord ainsi que le syst`eme de type Hodge-Laplace associ´e δ(A(x)dω)+dδω = f, avec les donn´ees au bord appropri´ees.Nous d´eduisons ´egalement sous la forme d’un corollaire l’existence et la r´egularit´e de solutions pour de syst`emesdu premier ordre de type div-rot. Nous d´eduisons ´egalement un r´esultat d’existence pour le probl`eme au limites semi-lin´eaire δ(A(x)(dω)) + f(ω)=λω in Ω, ν ∧ ω =0on∂Ω. Pour finir, nous discutons bri`evement des r´esultatsd’existence pour des op´erarteurs de Maxwell quasilin´eaires δ(A(x, dω)) = f, avec diff´erentes donn´ees au bord. Mots-cl´es Calcul des variations, formes diff´erentielles, quasiconvexit´e,probl`eme de mini- mization, semicontinuit´e, op´erateur de Maxwell. 2 Abstract In this thesis we study calculus of variations for differential forms. In the first part we develop the framework of direct methods of calculus of variations in the context of minimization problems for functionals of one or several differential forms of the type, f(dω), f(dω1,...,dωm)and f(dω, δω). Ω Ω Ω We introduce the appropriate convexity notions in each case, called ext. polyconvexity, ext. quasiconvexity and ext. one convexity for functionals of the type Ω f(dω), vectorial ext. poly- convexity, vectorial ext. quasiconvexity and vectorial ext. one convexity for functionals of the type f(dω1,...,dω )andext-int. polyconvexity, ext-int. quasiconvexity and ext-int. one Ω m convexity for functionals of the type Ω f(dω, δω). We study their interrelationships and the connections of these convexity notions with the classical notion of polyconvexity, quasiconvex- ity and rank one convexity in classical vectorial calculus of variations. We also study weak lower semicontinuity and weak continuity of these functionals in appropriate spaces, address coercivity issues and obtain existence theorems for minimization problems for functionals of one differential forms. In the second part we study different boundary value problems for linear, semilinear and quasilinear Maxwell type operator for differential forms. We study existence and derive interior regularity and L2 boundary regularity estimates for the linear Maxwell operator δ(A(x)dω)=f with different boundary conditions and the related Hodge Laplacian type system δ(A(x)dω)+dδω = f, with appropriate boundary data. We also deduce, as a corollary, some existence and regularity results for div-curl type first order systems. We also deduce existence results for semilinear boundary value problems δ(A(x)(dω)) + f(ω)=λω in Ω, ν ∧ ω =0on∂Ω. Lastly, we briefly discuss existence results for quasilinear Maxwell operator δ(A(x, dω)) = f, with different boundary data. Key words Calculus of variations, differential forms, quasiconvexity, minimization problem, semicontinuity, Maxwell operator. 2 Acknowledgements I have benefited immensely from the support and company of a number of people during the preparation of this thesis. I take this opportunity to express my sincere thanks and gratitude to my thesis director Bernard Dacorogna. Besides his constant encouragement, support and advice, our discussions in mathematics was instrumental in preparing this thesis. The second part of this thesis origi- nated with one of his questions about possible generalizations of div-curl type systems and his suggestion to try to generalize a result about semilinear Maxwell equations, to k-differential forms. The first part essentially grew out of the work I did in collaboration with him and Saugata Bandyopadhyay. Furthermore, his patience in reading a draft of this thesis and his numerous comments and suggestions were immensely helpful. I express my sincere thanks and gratitude to Saugata Bandyopadhyay. Not only did the first part grew out of our collaboration with Bernard Dacorogna, but Saugata and me continued to elaborate further on those ideas. A large portion of materials in the first part owes much to him. I had the luck of sharing my office with S´ebastienBasterrechea and to have Gyula Csat´oin theofficenexttome.TheimportanceoftheirfriendshiptothisthesisandmylifeinSwitzerland can not be overstated. It is also impossible to list in the numerous ways they have helped me. Without their company as friends, the mathematical discussions we had and their constant help in numerous little details of practical life in Switzerland, the years I spent preparing this thesis would have been very difficult. I thank my friends and colleagues Laura Keller, Linda de Cave and David Str¨utt and my friends Ludovic Karpincho and Souvik Das. Their company constituted a large part of my social life in Switzerland. I would also like to thank Virginie Ledouble, Anna Dietler and Catherine Risse for their constant help and support regarding virtually every academic and administrative matters in EPFL. I express my thanks to Jan Kristensen, Petru Mironescu and Ho`ai-Minh Nguyˆenfor agreeing to be the official referees of this thesis and also Joachim Krieger for agreeing to be the president of the jury committee. Of course, this thesis could never have been written without the unconditional love and support of my family and friends. 1 Contents 1 Introduction 5 1.1Analysiswithdifferentialforms............................ 5 1.2FunctionalAnalyticsetting.............................. 6 1.3Classicalcalculusofvariations............................. 7 1.4Calculusofvariationsfordifferentialforms...................... 7 1.5Functionsofexteriorderivativeofasingledifferentialform............ 8 1.6Functionsofexteriorderivativesofseveraldifferentialforms............ 13 1.7Functionsofexteriorandinteriorderivativeofasingledifferentialform..... 17 1.8Relationshipwiththeclassicalcalculusofvariations................ 19 1.9Maxwelloperator.................................... 23 1.10Organization...................................... 29 2 Differential forms 30 2.1Introduction....................................... 30 2.2Exteriorforms..................................... 31 2.2.1 Definitionsandmainproperties........................ 31 2.2.2 Divisibility ................................... 33 2.3Differentialformsandtheirderivatives........................ 34 2.4 Function Spaces of differential forms on Rn ..................... 35 2.4.1 Partly Sobolev classes ............................. 35 2.4.2 TraceonpartialSobolevspaces........................ 36 2.4.3 GaffneyinequalityandHarmonicfields................... 40 2.5Decompositiontheoremsandconsequences...................... 42 2.5.1 Hodge-Morreydecomposition......................... 42 2.5.2 Classicalboundaryvalueproblemsfordifferentialforms.......... 44 2.5.3 ImportantConsequences............................ 50 I Direct Methods in Calculus of Variations for Differential Forms 54 3 Functionals depending on exterior derivative of a single differential form 56 3.1Introduction....................................... 56 3.2NotionsofConvexity.................................. 57 3.2.1 Definitions................................... 57 3.2.2 Preliminarylemmas.............................. 60 2 3.2.3 Mainproperties................................. 65 3.3Thequasiaffinecase.................................. 70 3.3.1 Somepreliminaryresults............................ 70 3.3.2 Thecharacterizationtheorem......................... 73 3.4Examples........................................ 79 3.4.1