Springer Monographs in Mathematics D. E. Edmunds W. D. Evans Elliptic Differential Operators and Spectral Analysis Springer Monographs in Mathematics Editors-in-chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. 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Desmond Evans (cid:129) Elliptic Differential Operators and Spectral Analysis 123 DavidE. Edmunds W.Desmond Evans Department ofMathematics Schoolof Mathematics University of Sussex University of Cardiff Brighton, UK Cardiff, UK ISSN 1439-7382 ISSN 2196-9922 (electronic) SpringerMonographs inMathematics ISBN978-3-030-02124-5 ISBN978-3-030-02125-2 (eBook) https://doi.org/10.1007/978-3-030-02125-2 LibraryofCongressControlNumber:2018960175 MathematicsSubjectClassification(2010): 35Jxx,35Pxx,35Qxx,47A10 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface In 1987, our book Spectral Theory and Differential Operators was published. It emphasisedthesymbioticrelationship,aswesawit,betweenthetheoryofcompact linear operators acting between Banach spaces and the study of boundary value problems for elliptic partial differential equations. Although there have been many advances in the theory since then, the book contained much material of lasting importance that is relatively unaffected by recent progress. Nevertheless, we feel thatitisnowappropriatetogiveanaccountofsometopics,botholdandnew,not coveredinthepreviousbook.Thecentralthemeisthatofellipticoperators:thisisa huge subject, well able to support many variations in approach. Those given here reflect our interests and limitations of our knowledge. ThereaderisassumedtobefamiliarwiththetheoryofLebesgueintegrationand basic facts concerning functional analysis and the theory of function spaces. A summary of the most fundamental results in these topics is provided in Chap. 1 together with references to more detailed accounts. With an eye to later usage, a Banach space version of the Lax-Milgram lemma is established, as is the deter- mination (due to Pichorides) of theexactvalue of the norm of the Riesz transform as a map from Lp to itself. The next two chapters cover well-known results involving the Laplace operator, such as maximum principles, Weyl’s lemma and thePerronapproachtotheDirichletproblem,togetherwithmaximumprinciplesfor second-orderellipticoperators.Chapter4studiestheclassicalDirichletproblemfor second-order elliptic operators and presents such familiar matters as Kellogg’s theorem and the Schauder boundary estimates. The approach given is that due to M. König in a series of papers and does not seem to have appeared in book form before now: we believe that it has pedagogical advantages over more traditional methods in being simpler and more direct. The next chapter provides various notions of ellipticity for operators of arbitrary order and establishes Gårding’s inequalityforuniformlystronglyellipticoperators. TheDirichletproblemfor such operatorsisdiscussed:toprovetheexistenceofaclassicalsolutionwouldentailthe establishmentofahigher-orderversionoftheSchauderestimates,andasweshrink from this formidable task, it is shown that a weak solution exists. Such solutions belong to Sobolev spaces based on the Hilbert space L ; and regularity theory is 2 v vi Preface neededtoshowthattheweaksolutionisactuallyclassical:indicationsofhowsuch argumentsgoaregivenforthePoissonequation.TheDirichleteigenvectorsofthe Laplacian are next considered: Courant’s min-max principle is proved, as are the analyticity ofeigenvectorsand theFaber–Krahn inequalityfor thefirsteigenvalue. The chapter concludes with a brief discussion of semigroup theory and its con- nection with spectral independence. Chapter6continuestheHilbertspaceapproachofthelastchapterandisdevoted to self-adjoint extensions of symmetric operators acting in a Hilbert space. After a brief resumé of the von Neumann theory, characterisations of self-adjoint exten- sions in terms of linear relations and boundary triplets are given, and associated gammafieldsandWeylM-functionsareintroducedandtheirmainpropertiesnoted; a comprehensive treatment may be found in the book by Schmüdgen [190]. The mainthemeofChap.6isanaccountoftheKrein-Vishik-Birmantheoryconcerning the positive self-adjoint extensions of positive symmetric operators, and Grubb’s extension of the theory to adjoint pairs of closed operators. Analogous results of Arlinski and his co-authors on the m-sectorial extensions of sectorial operators are also included. Application of these abstract results is made in the next chapter to realisationsofsecond-orderellipticoperators.ThefirstfoursectionsofChap.7deal with symmetric Sturm–Liouville operators satisfying minimal assumptions, fol- lowed by a brief description for coercive sectorial operators of Sturm–Liouville type.Then,anoutlineisgivenoftheworkofGrubbinwhichherabstracttheoryfor determiningalltheclosedrealisationsofanadjointpairofoperatorsA; A0isapplied to uniformly elliptic operators generated by differential expressions Awith smooth coefficients and formal adjoint A0, defined on a smooth domain X(cid:2)Rn ðn(cid:3)2Þ: ThisleadstotheidentificationofallclosedrealisationsofAbymeansofboundary conditions on @X expressed in terms of differential operators acting between function spaces defined on @X: Chapter 8 marks a break with the Hilbert space approach: the necessity for this stems from consideration of the Poisson equation with right-hand side f that belongs to Lp for some p2ð1;2Þ but does not belong to L2: The methods of Chap.5arethennotapplicable,butitturnsoutthattheexistenceofanappropriate type of weak solution in the context of Lp can be established. To do this, some techniques introduced by Simader and Sohr can be used: we give a simplified version of their approach adapted to the particular case we consider. Chapter 9 is devoted to the p(cid:4)Laplacian, the literature on which is so enormous as to appear overwhelming to those unfamiliar with the topic. Our object here is modest: by concentrating on a smallnumber of problems, we hope to encourage thenovice to pluckupenoughcouragetoventuremoredeeplyintothesubject.Theexistenceof asolutionofthecorrespondingDirichletproblemisproved,togetherwithavariety ofresultsconcerningeigenvalues,includingaversionoftheCourantnodaldomain theorem.Thefinalthreechaptersareintendedtogivesomeideaofcurrentworkin whichweareinterested.Chapter10describessomeveryrecentformsoftheRellich inequality; Chap. 11 provides further properties of Sobolev embeddings, such as necessary and sufficient conditions for a Sobolev embedding to be nuclear, and a Preface vii characterisation of the subspace of a Sobolev space consisting of functions with zerotrace;Chap.12discussespositiveoperatorswhichmodelrelativisticproperties of the Dirac operator, special attention being paid to the self-adjoint realisations of the Brown–Ravenhall operator defined on a domain X$R3. Notes are given at the end of most chapters to provide the reader with further references and indica- tions of directions taken by current research. Itishopedthatthecocktailofresultsandtechniquespresentedherewithwhich diverse questions related to the spectral theory of differential operators may be attackedwill prove to be of interest, especially since a good deal ofthe material is hard to find in book form. Chapters are divided into sections, and some sections are divided into subsec- tions. Theorems, corollaries, lemmas, propositions, remarks and equations are numbered consecutively in each section. Brighton, UK David E. Edmunds Cardiff, UK W. Desmond Evans Contents 1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Spaces of Continuous Functions . . . . . . . . . . . . . . . . . 9 1.3.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 The Hilbert and Riesz Transforms . . . . . . . . . . . . . . . . . . . . . . 24 2 The Laplace Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1 Mean Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Representation of Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Dirichlet Problems: The Method of Perron. . . . . . . . . . . . . . . . 51 2.4 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3 Second-Order Elliptic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Maximum Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 The Classical Dirichlet Problem for Second-Order Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 The Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 More General Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Elliptic Operators of Arbitrary Order . . . . . . . . . . . . . . . . . . . . . . 83 5.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Gårding’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 The Dirichlet Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4 A Little Regularity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5 Eigenvalues of the Laplacian. . . . . . . . . . . . . . . . . . . . . . . . . . 99 ix x Contents 5.6 Spectral Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 Operators and Quadratic Forms in Hilbert Space . . . . . . . . . . . . . 115 6.1 Self-Adjoint Extensions of Symmetric Operators . . . . . . . . . . . 115 6.2 Characterisations of Self-Adjoint Extensions. . . . . . . . . . . . . . . 121 6.2.1 Linear Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2.2 Boundary Triplets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2.3 Gamma Fields and Weyl Functions. . . . . . . . . . . . . . . 128 6.3 The Friedrichs Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.4 The Krein-Vishik-Birman (KVB) Theory . . . . . . . . . . . . . . . . . 136 6.5 Adjoint Pairs and Closed Extensions . . . . . . . . . . . . . . . . . . . . 144 6.6 Sectorial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.6.1 The Friedrichs Extension . . . . . . . . . . . . . . . . . . . . . . 153 6.6.2 The Krein-von Neumann Extension. . . . . . . . . . . . . . . 156 6.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7 Realisations of Second-Order Linear Elliptic Operators . . . . . . . . . 159 7.1 Sturm–Liouville Operators: Basic Theory. . . . . . . . . . . . . . . . . 159 7.1.1 The Regular Problem . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.1.2 One Singular Point. . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.1.3 Two Singular End-Points . . . . . . . . . . . . . . . . . . . . . . 165 7.1.4 The Titchmarsh–Weyl Function and Spectrum. . . . . . . 166 7.2 KVB Theory for Positive Sturm–Liouville Operators . . . . . . . . 168 7.2.1 Semi-boundedness and Oscillation Theory. . . . . . . . . . 168 7.2.2 Kalf’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.3 Application of the KVB Theory. . . . . . . . . . . . . . . . . . . . . . . . 175 7.3.1 The Limit-Point Case at b. . . . . . . . . . . . . . . . . . . . . . 175 7.3.2 The Case of b Regular or Limit Circle, and ¿u¼0 Non-oscillatory at b . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3.3 Limit-Point and Limit-Circle Criteria. . . . . . . . . . . . . . 183 7.4 Coercive Sectorial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.4.1 The Case dim(ker T*)¼2. . . . . . . . . . . . . . . . . . . . . . 188 7.5 Realisations of Second-Order Elliptic Operators on Domains. . . 188 7.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8 The Lp Approach to the Laplace Operator . . . . . . . . . . . . . . . . . . . 203 8.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.3 Existence of a Weak Lp Solution. . . . . . . . . . . . . . . . . . . . . . . 207 8.4 Other Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211