DOCSLIB.ORG

On Non-Local Boundary Value Problems for Elliptic Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Non-Local Boundary Value Problems for Elliptic Operators On Non-local Boundary Value Problems for Elliptic Operators Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atzu K¨oln vorgelegt von Christian Frey aus Bremen K¨oln,2005 Berichterstatter: Prof. Dr. Matthias Lesch Prof. Dr. Horst Lange Tag der m¨undlichen Pr¨ufung: 13. Juli 2005 Meinen Eltern gewidmet Contents Zusammenfassung 5 Abstract 7 Introduction 9 0 Preliminaries and General Assumptions 13 1 Boundary Value Problems 19 1.1 Weak Traces for Elliptic Operators . 20 1.2 Basic Properties of Boundary Conditions . 26 1.3 Functional Analysis for Boundary Value Problems . 36 2 Regularity and Well-posedness 41 2.1 The Regularity Condition . 43 2.2 Higher Regularity . 51 2.3 The Calder´onProjection . 54 2.4 Bojarski’s Theorem and the Agranoviˇc-Dynin-Formula . 63 2.5 The Formally Self-adjoint Case . 66 2.6 Perturbation Theory for Well-posed Boundary Problems . 70 3 Operators of Dirac type 73 3.1 Dirac Bundles on Manifolds with Boundary . 73 3.1.1 The Gauß-Bonnet and Signature Operator . 76 3.1.2 The Spin Dirac Operator . 79 3.2 The Calder´onProjection of a First Order Elliptic Operator . 82 3.3 The Calder´onProjection of a Dirac Operator . 84 3.4 Cobordism Invariance of the Index . 86 4 Operators of Laplace Type 89 4.1 Reduction to the Model Cylinder . 91 4.2 Regularity and Well-posedness . 95 4.3 Examples of Boundary Value Problems for Laplacians . 98 4.3.1 Dirichlet and Neumann Boundary Conditions . 98 4.3.2 Rotating Vector Fields for Surfaces with Boundary: A Theorem of I. N. Vekua . 100 4.3.3 Absolute and relative boundary condition . 105 4.4 The Dirichlet-to-Neumann Operator . 108 4.5 Self-adjoint Realisations and Semi-boundedness . 111 3 4 4.5.1 A Characterisation of Well-posed Self-adjoint Realisations and Canonical Forms of Boundary Conditions . 111 4.5.2 Criteria for Lower Boundedness . 116 A Functional Analysis 121 A.1 Fredholm Theory . 121 A.2 The Spectral Invariance of a Certain Spectral Triple . 125 Bibliography 131 Zusammenfassung In dieser Arbeit werden nicht-lokale Randwertprobleme f¨ur elliptische Differentialopera- toren auf kompakten Mannigfaltigkeiten mit glattem Rand beleuchtet. Dabei wird zun¨achst die Randwerttheorie f¨ur allgemeine elliptische Differentialoperatoren aufge- baut. U.a. wird bewiesen, dass alle schwachen L¨osungen, die einer gegebenen Randbe- dingung gen¨ugen, genau dann stark sind, wenn f¨urdie starken L¨osungeneine gewisse Regularit¨atsabsch¨atzung gilt. Die hier betrachteten Randbedingungen lassen sich durch Projektionen beschreiben, die folgende Bedingung erf¨ullen. Der Operator und sein Kommutator mit einem Erzeuger der Sobolev-Skala auf dem Rand ist auf der gesamten Sobolev-Skala beschr¨ankt. Als Erzeuger kann ein beliebiger elliptischer Pseudodifferen- tialoperator mit skalarem Hauptsymbol gew¨ahltwerden. Im Vergleich zur bestehenden Literatur ist diese Klasse von Randbedingungen sehr allgemein und umfasst sowohl spektrale Randbedingungen als auch einige nicht-pseudolokale Randbedingungen. Wie im Anhang bewiesen wird, bilden die Operatoren, die obiger Kommutator-Bedingung gen¨ugen,eine lokale C∗-Algebra, was sich als grundlegend f¨ur den gesamten Aufbau der Theorie herausstellt. Im Mittelpunkt der Regularit¨atstheorie steht der Calder´on-Projektor bzw. sein Bild, der Cauchy-Daten-Raum Λ0. Wir geben hier einen vereinfachten Beweis daf¨ur, dass der Calder´on-Projektor eines beliebigen elliptischen Differentialoperators ein klassischer Pseudodifferentialoperator auf dem Rand ist. Zu einer Randbedingung betrachten wir den Raum aller zul¨assigen Randwerte Λ. Die wichtigsten Eigenschaften einer Fortset- zung, Regularit¨at,Korrektgestelltheit und Selbstadjungiertheit, lassen sich dann ¨uber das Raum-Paar (Λ, Λ0) beschreiben. Im Falle pseudodifferentieller Randbedingungen gelangen wir so zu den bekannten Kriterien. F¨uralle formal selbst-adjungierten Differen- tialoperatoren wird gezeigt, dass die Gesamtheit der regul¨arenselbstadjungierten Fort- setzungen durch die zu Λ0 assoziierte Fredholm-Langrange Grassmannsche parametri- siert wird. Danach wenden wir die allgemeinen Ergebnisse auf zwei der wichtigsten Klassen geo- metrischer Differentialoperatoren, Dirac- und Laplace-Operatoren, an. F¨ur eine Reihe von Dirac-Operatoren zeigen wir, wie das Hauptsymbol des Calder´on-Projektors aus dem tangentialen Anteil des Dirac-Operators bestimmt werden kann. Auf diese Weise erhalten wir die Regularit¨atstheorie von Randwertproblemen f¨ur den Hodge-de Rham-, Signatur-, Gauß-Bonnet- und den Spin-Dirac-Operator. Die Invarianz des Index eines Dirac-Operators unter Kobordismen wird mit Hilfe elementarer Fredholmtheorie aus den Eigenschaften des Cauchy-Daten-Raums gefolgert. F¨ur Laplace-Operatoren besprechen wir eine Reihe klassischer Randwertprobleme. Es werden die Randbedingungen, die zu selbstadjungierten Fortsetzungen f¨uhren, ge- nauer charakterisiert. Weiter wird ein hinreichendes und notwendiges Kriterium f¨ur die Halbbeschr¨anktheit selbstadjungierter Fortsetzungen von Laplace-Operatoren be- wiesen. 5 6 Abstract In this thesis we discuss boundary value problems for elliptic differential operators where the boundary conditions belong to a certain class of non-local operators. More precisely, the boundary conditions are given in terms of the standard traces and operators with closed range on the boundary satisfying the following condition: The operator and its commutator with some fixed generator of the Sobolev scale are bounded on the full Sobolev scale of the boundary. This generator is an arbitrary first order elliptic pseudodifferential operator with scalar principal symbol. It is shown that this classe operators form a local C∗-algebra which contains all pseudodifferential operators with closed range. We establish a regularity theory characterising all boundary conditions which give rise to regular, well-posed and self-adjoint extensions. We show that for a given bound- ary condition a certain regularity estimate is equivalent to the fact that all weak solu- tions satisfying this condition are strong ones. The theory is based on the properties of the Cauchy data space, Λ0, and on the notion of Fredholm pairs. We construct the Calder´onprojection for any elliptic differential operator avoiding the unique contin- uation principle. We prove that our construction yields a classical pseudodifferential operator. It follows that the Fredholm-Lagrange-Grassmannian w.r.t. Λ0 is non-empty and parametrises the space of all regular self-adjoint extensions of a formally self-adjoint operator. Using only elementary Fredholm theory we deduce cobordism invariance for the index of Dirac operators from the facts that the Cauchy data space is Lagrangian, on the one hand, and that the Calder´onprojection differs from the positive spectral projection merely by a compact operator, on the other. Many classical examples of Dirac and Laplace operators are revisited. Moreover, for Laplace operators we describe all regular self-adjoint extensions in terms of boundary conditions and establish a necessary and sufficient criterion for semi-boundedness. 7 8 Introduction Boundary value problems are found in many mathematical models of physical systems. Among these are classical mechanical systems such as vibrating strings or clamped plates as well as quantum mechanical systems, e.g. particles in a box and quantum hall systems. Typically, an evolution process, e.g. diffusion or wave propagation, is modelled by a partial differential equation on a manifold with boundary together with a boundary condition. Whereas the partial differential equation models how temperature diffuses or how waves propagate in the interior of a medium, the boundary condition tells us e.g. if the temperature is fixed at the edge, if the boundary is an ideal isolator, or how waves are reflected. Many of the above models contain as its main ingredient a linear elliptic differential operator, so that the question of boundary conditions for these operator naturally arises. However, it should be emphasised that elliptic boundary value problems, e.g. for Dirac operators, are studied in its own right by mathematicians and have important applications in geometry and topology. In spite of their long history and outstanding role in mathematics and physics, it is hard to find literature on elliptic boundary value problems, that does not make any special assumptions on the shape of the operators and boundary condition. Certainly, the bigger part of the above-mentioned examples does not justify to study general elliptic boundary value problems the more so as natural restrictions on the elliptic operator allow us to employ powerful techniques, e.g. by treating the partial differential equation as a variational problem. Nevertheless, I am convinced that the fundamental questions that one is concerned with when studying an elliptic boundary value problem can be treated in a completely general framework. Indeed, there are by now successful approaches that aim at a generalisation of the theory of pseudodifferential operators on manifolds without boundary, the most promi- nent being the early work of L. Boutet de Monvel ([BdM71]). However, the mathematics involved in these theories seems rather complicated and thus out of reach for many non- specialists. In order to bridge this gap the first part of the thesis (Chapter 1 and 2) aims at a description of all boundary value problems having certain regularity and bounded- ness properties that are essential for most applications. The main theorems proved in these chapters have a long
Load more

Recommended publications
  • On Elliptic Partial Differential Equations Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 13, No 2 (1959), P
    ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze L. NIRENBERG On elliptic partial differential equations Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 13, no 2 (1959), p. 115-162 <http://www.numdam.org/item?id=ASNSP_1959_3_13_2_115_0> © Scuola Normale Superiore, Pisa, 1959, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS by L. NIRENBERG (New York) (*) Outline. This series of lectures will tonch on a number of topics in the theory of elliptic differential equations. Iii Lecture I we discuss the fundamental solution for equations with constant coefficients. Lecture .2 is concerned with Calculus inequalities including the well known oues of Sobolev. In le- ctures 3 and 4 we present the Hilbert space approach to the Dirichlet pro- blem for strongly elliptic systems, and describe various inequalities. Lectures 5 and 6 comprise a self contained proof of the well known fact that « weak » solutions of elliptic equations with sufficiently « smooth &#x3E;&#x3E; coefacients are classical solutions. In Lectures 7 and 8 we describe some work of Agmoii, Douglis, Nirenberg [14] concerning estimates near the boundary for solutions of elliptic equations satisfying boundary conditions.
    [Show full text]
  • Elliptic Boundary Problems
    CHAPTER 8 Elliptic boundary problems Summary Elliptic boundary problems are discussed, especially for operators of Dirac type. We start with a discussion of manifolds with boundary, including functions spaces and distributions. This leads to the `jumps formula' for the relationship of the action of a differential operator to the operation of cutting off at the boundary; this is really Green's formula. The idea behind Calder`on'sapproach to boundary problems is introduced in the restricted context of a dividing hypersurface in a manifold without boundary. This includes the fundamental result on the boundary behaviour of a pseudodifferential operator with a rational symbol. These ideas are then extended to the case of an operator of Dirac type on a compact manifold with boundary with the use of left and right parametrices to define the Calder`on projector. General boundary problems defined by pseudodifferential projections are discussed by reference to the ‘Calder`onrealization' of the operator. Local boundary conditions, and the corresponding ellipticity conditions, are then discussed and the special case of Hodge theory on a compact manifold with boundary is analysed in detail for absolute and relative boundary conditions. Introduction Elliptic boundary problems arise from the fact that elliptic differential operators on compact manifolds with boundary have infinite dimensional null spaces. The main task we carry out below is the parameterization of this null space, in terms of boundary values, of an elliptic differential operator on a manifold with boundary. For simplicity of presentation the discussion of elliptic boundary problems here will be largely confined to the case of first order systems of differential operators of Dirac type.
    [Show full text]
  • The Lp Resolvents of Elliptic Operators with Uniformly Continuous Coefficients
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Differential Equations 188 (2003) 555–568 The Lp resolvents of elliptic operators with uniformly continuous coefficients Yoichi Miyazaki*,1 School of Dentistry, Nihon University, 8-13, Kanda-Surugadai 1-chome, Chiyoda-ku, Tokyo 101-8310, Japan Received September 19, 2001; revised June 13, 2002 Abstract We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1opoN; A is a bounded linear operator from the Lp Sobolev space Hm;p into HÀm;p: We will prove that ðA À lÞÀ1 exists in HÀm;p for some l and estimate its operator norm. r 2002 Elsevier Science (USA). All rights reserved. Keywords: Lp theory; Elliptic operator; Non-smooth coefficient; Divergence form; Resolvent; Uniformly continuous 1. Introduction Let us consider the elliptic operator defined in the n-dimensional Euclidean space Rn in the divergence form X a b AuðxÞ¼ D ðaabðxÞD uðxÞÞ; ð1:1Þ jaj;jbjpm n where x ¼ðx1; y; xnÞ is a generic point in R and we use the notations pffiffiffiffiffiffiffi a a1 ? an D ¼ D1 Dn ; Dj ¼ À1@=@xj *Corresponding author. E-mail address: [email protected]. 1 Supported partly by Sato Fund, Nihon University School of Dentistry. 0022-0396/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 0396(02)00109-2 556 Y.
    [Show full text]
  • Remarks on Regularity for $ P $-Laplacian Type Equations in Non
    REMARKS ON REGULARITY FOR p-LAPLACIAN TYPE EQUATIONS IN NON-DIVERGENCE FORM AMAL ATTOUCHI AND EERO RUOSTEENOJA Abstract. We study a singular or degenerate equation in non-divergence form modeled on the p-Laplacian, γ N −|Du| ∆u +(p − 2)∆∞u = f in Ω. We investigate local C1,α regularity of viscosity solutions in the full range γ > −1 and p > 1, and provide local W 2,2 estimates in the restricted cases where p is close to 2 and γ is close to 0. 1. Introduction In this paper we study local regularity properties of solutions of the fol- lowing nonlinear PDE, Du γ∆N u := Du γ ∆u + (p 2)∆N u = f in Ω, (1.1) − | | p −| | − ∞ where γ ( 1, ), p (1, ), f C(Ω), Ω Rn with n 2, and ∈ − ∞ ∈ ∞ ∈ ⊂ ≥ ∆N u := D2u Du , Du is the normalized infinity Laplacian. Equation ∞ h |Du| |Du|i (1.1) is degenerate when γ > 0 and singular when 1 < γ 0, and it is appropriate to use the concept of viscosity solution. The− operator≤ Du γ∆N u | | p is a generalization of the standard p-Laplacian ∆ u := div ( Du p−2Du). As p | | special cases we mention the variational p-Laplace equation ∆pu = f when − N γ = p 2, and the non-variational normalized p-Laplace equation ∆p u = f when −γ = 0. − Non-variational operators modeled by the p-Laplacian have gained in- arXiv:1710.07506v2 [math.AP] 2 Oct 2018 creasing interest during the last 15 years. For C2 domains with W 2,∞ boundary data, Birindelli and Demengel [9, 10, 12] showed global H¨older and local Lipschitz estimates for a class of fully nonlinear elliptic equations including equation (1.1) by using Ishii-Lions’ method.
    [Show full text]
  • The Resolvent Parametrix of the General Elliptic Linear Differential Operator: a Closed Form for the Intrinsic Symbol
    transactions of the american mathematical society Volume 310, Number 2, December 1988 THE RESOLVENT PARAMETRIX OF THE GENERAL ELLIPTIC LINEAR DIFFERENTIAL OPERATOR: A CLOSED FORM FOR THE INTRINSIC SYMBOL S. A. FULLING AND G. KENNEDY ABSTRACT. Nonrecursive, explicit expressions are obtained for the term of arbitrary order in the asymptotic expansion of the intrinsic symbol of a resol- vent parametrix of an elliptic linear differential operator, of arbitrary order and algebraic structure, which acts on sections of a vector bundle over a manifold. Results for the conventional symbol are included as a special case. 1. Introduction. As is well known, the resolvent operator, (A - A)-1, plays a central role in the functional analysis associated with an elliptic linear differential operator A. In particular, from it one can easily obtain the corresponding heat operator, e~tA, for t G R+ and semibounded A. Furthermore, detailed knowledge of the terms in the asymptotic expansions of the integral kernels of the resolvent and heat operators is of great value in calculating the asymptotics of eigenvalues and spectral functions [12, 25, 2, 3, 33]; partial solutions of inverse problems [26, 15]; indices of Fredholm operators [1, 19, 20]; and various physical quantities, including specific heats [4], partition functions [49], renormalized effective actions [38, 14, 34, 45, 8], and renormalized energy-momentum tensors [9, 43, 44]. (The references given here are merely representative.) It is important, therefore, to have available an efficient and general method for calculating such terms. The present paper addresses this topic for the resolvent operator, or, more precisely, a resolvent parametrix; later papers will treat the heat operator.
    [Show full text]
  • ELLIPTIC COMPLEXES and INDEX THEORY 1. Sobolev Theory on R
    ELLIPTIC COMPLEXES AND INDEX THEORY VISHWAMBHAR PATI 1. Sobolev Theory on Rn As general references for this section, see the books [Nar], Ch.3, [Gil], Ch.1, [Hor], Ch I, II and [Rud], Ch. 6,7. 1.1. Test functions and distributions. We introduce some standard notation. For a multi-index α = (α1; ::; αn) of length n, the symbol α := i αi, and α! := α1!; ::; αn!. For derivatives, we denote: j j P α1 αn α @ @ @ 1 dx = :::::: ; @j = ; Dj = @j @x1 @xn @xj p 1 − α1 α2 αn α Finally x1 x2 :::xn will be denoted simply by x . We can define some standard function spaces on Rn. For us, functions will always be complex valued. Definition 1.1.1 (Standard function spaces on Rn). The function spaces defined below are all complex vector spaces, and to define a topology on them, it is enough to define convergence to zero. (i): n n C1(R ) = smooth functions on R f g α Define fn 0 if dx fn 0 uniformly on compact sets for all α 0. This space is also sometimes denoted !by analysts. ! j |≥ E (ii): Rn Rn α C01( ) = f C1( ) : lim dx f = 0 for all α 0 x f 2 !1 j |≥ g This is the space of all smooth functions whose derivatives of all orders vanish at infinity. The topology n in this space is the subspace topology from C1(R ). It is often denoted . E0 (iii): n n C1(R ) = f C1(R ) : support f is compact c f 2 g Its topology is defined by fn 0 if there exists a compact set K such that suppfn K for all n and α ! ⊂ n d f 0 uniformly on K for all α 0.
    [Show full text]
  • 1. Harmonic Functions 2. Perron's Method 3. Potential Theor
    Elliptic and Parabolic Equations by J. Hulshof Elliptic equations: 1. Harmonic functions 2. Perron’s method 3. Potential theory 4. Existence results; the method of sub- and supersolutions 5. Classical maximum principles for elliptic equations 6. More regularity, Schauder’s theory for general elliptic operators 7. The weak solution approach in one space dimension 8. Eigenfunctions for the Sturm-Liouville problem 9. Generalization to more dimensions Parabolic equations: 10. Maximum principles for parabolic equations 11. Potential theory and existence results 12. Asymptotic behaviour of solutions to the semilinear heat equation Functional Analysis: A. Banach spaces B. Hilbert spaces C. Continous semigroups and Liapounov functionals 1 1. Harmonic functions Throughout this section, Ω ⊂ Rn is a bounded domain. 1.1 Definition A function u ∈ C2(Ω) is called subharmonic if ∆u ≥ 0 in Ω, harmonic if ∆u ≡ 0 in Ω, and superharmonic if ∆u ≤ 0 in Ω. 1.2 Notation The measure of the unit ball in Rn is Z n/2 n 2 2 2π ωn = |B1| = |{x ∈ R : x1 + ... + xn ≤ 1}| = dx = . B1 nΓ(n/2) The (n − 1)-dimensional measure of the boundary ∂B1 of B1 is equal to nωn. 1.3 Mean Value Theorem Let u ∈ C2(Ω) be subharmonic, and n BR(y) = {x ∈ R : |x − y| ≤ R} ⊂ Ω. Then 1 Z u(y) ≤ n−1 u(x)dS(x), nωnR ∂BR(y) where dS is the (n − 1)-dimensional surface element on ∂BR(y). Also 1 Z u(y) ≤ n u(x)dx. ωnR BR(y) Equalities hold if u is harmonic. Proof We may assume y = 0.
    [Show full text]
  • Elementary Theory and Methods for Elliptic Partial Differential Equations
    Elementary Theory and Methods for Elliptic Partial Differential Equations John Villavert Contents 1 Introduction and Basic Theory 4 1.1 Harmonic Functions . 5 1.1.1 Mean Value Properties . 5 1.1.2 Sub-harmonic and Super-harmonic Functions . 8 1.1.3 Further Properties of Harmonic Functions . 11 1.1.4 Energy and Comparison Methods for Harmonic Functions . 14 1.2 Classical Maximum Principles . 17 1.2.1 The Weak Maximum Principle . 17 1.2.2 The Strong Maximum Principle . 18 1.3 Newtonian and Riesz Potentials . 21 1.3.1 The Newtonian Potential and Green's Formula . 21 1.3.2 Riesz Potentials and the Hardy-Littlewood-Sobolev Inequalitiy . 23 1.3.3 Green's Function and Representation Formulas of Solutions . 25 1.3.4 Green's Function for a Half-Space . 26 1.3.5 Green's Function for a Ball . 28 1.4 H¨olderRegularity for Poisson's Equation . 31 1.4.1 The Dirichlet Problem for Poisson's Equation . 33 1.4.2 Interior H¨olderEstimates for Second Derivatives . 36 1.4.3 Boundary H¨olderEstimates for Second Derivatives . 40 2 Existence Theory 43 2.1 The Lax-Milgram Theorem . 43 2.1.1 Existence of Weak Solutions . 44 2.2 The Fredholm Alternative . 47 2.2.1 Existence of Weak Solutions . 48 2.3 Eigenvalues and Eigenfunctions . 52 1 2.4 Topological Fixed Point Theorems . 53 2.4.1 Brouwer's Fixed Point Theorem . 54 2.4.2 Schauder's Fixed Point Theorem . 55 2.4.3 Schaefer's Fixed Point Theorem . 56 2.4.4 Application to Nonlinear Elliptic Boundary Value Problems .
    [Show full text]
  • Lecture Notes Elliptic Equations
    Lecture Notes Elliptic Equations Course held by Notes written by Prof. Antonio Tarsia Francesco Paolo Maiale Department of Mathematics Pisa University February 7, 2018 Disclaimer These incomplete notes came out of the Elliptic Equations course, held by Professor Antonio Tarsia in the second semester of the academic year 2016/2017. I have used them to study for the exam; hence they have been reviewed thoroughly. Unfortunately, there may still be many mistakes and oversights; to report them, send me an email at francescopaolo (dot) maiale (at) gmail (dot) com. Contents 1 Introduction To Elliptic Problems4 1.1 Ellipticity Condition................................4 1.2 Elliptic Systems...................................8 1.3 Definitions of Solution............................... 10 1.4 Boundary Problems................................ 11 1.5 The Laplacian Operator and The Calculus of Variations............ 14 2 Existence Results for Elliptic Problems 16 2.1 Near Operators Theory.............................. 16 2.2 Historical Digression on Near Operators..................... 19 2.3 Regularity Conditions............................... 23 2.4 Lax-Milgram Theorem............................... 28 2.5 Garding Inequality................................. 29 2.6 Dirichlet problem for linear systems....................... 32 2.7 Global Existence via Special Operators...................... 35 3 Regularity in Sobolev Spaces 38 3.1 Nirenberg Lemmas................................. 38 3.2 Interior Regularity Theory............................. 41 3.3
    [Show full text]
  • The Invariance of the Index of Elliptic Operators
    The Invariance of the Index of Elliptic Operators Constantine Caramanis∗ Harvard University April 5, 1999 Abstract In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem, which states, among other things, that the space of elliptic pseudodifferential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under sufficiently small perturbations. By developing the machinery of distributions and in particular Sobolev spaces, this paper addresses this more specific part of the famous Theorem from a completely analytic approach. We first prove the regularity of elliptic operators, then the finite dimensionality of the kernel and cokernel, and finally the invariance of the index under small perturbations. ∗[email protected] 1 Acknowledgements I would like to express my thanks to a number of individuals for their con- tributions to this thesis, and to my development as a student of mathematics. First, I would like to thank Professor Clifford Taubes for advising my thesis, and for the many hours he spent providing both guidance and encouragement. I am also indebted to him for helping me realize that there is no analysis without geometry. I would also like to thank Spiro Karigiannis for his very helpful criti- cal reading of the manuscript, and Samuel Grushevsky and Greg Landweber for insightful guidance along the way. I would also like to thank Professor Kamal Khuri-Makdisi who instilled in me a love for mathematics. Studying with him has had a lasting influence on my thinking.
    [Show full text]
  • NON-LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 1. The
    NON-LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS PIOTR HAJLASZ 1. The classical Dirichlet problem and the origin of Sobolev spaces The classical Dirichlet problem reads as follows. Given an open domain n 0 2 0 Ω ⊂ R and g ∈ C (∂Ω). Find u ∈ C (Ω) ∩ C (Ω) such that −∆u = 0 in Ω , (1.1) u|∂Ω = g . This problem originates from the XIX’th century physics. To begin with, we shall provide a brief explanation of the physical context of the equation. This will also provide us with an intuition for the principles of the calculus of variations that will be developed later. 3 3 Let Ω ⊂ R be a vacuum region and let E :Ω → R be an electric field (electrostatic case). Given two points x, y ∈ Ω, the integral Z y (1.2) E · ds x does not depend on the choice of the curve that joins x with y inside Ω (provided Ω is simply connected). By the definition, (1.2) equals to Z b E(γ(t)) · γ˙ (t) dt, a where γ is a parametrization of the given curve that joins x and y. Here and in the sequel A·B denotes the scalar product. We will also denote the scalar product by hA, Bi. Fix x0 ∈ Ω and define the potential u as follows Z x u(x) = − E · ds. x0 The potential u is a scalar function defined up to a constant (since we can change the base point x0). We have E = −∇u . Date: June 23, 2010. 1 2 PIOTR HAJLASZ It is well known that the electric field is divergence free, div E = 0, and hence ∆u = div ∇u = −div E = 0.
    [Show full text]
  • Symmetry and Non-Uniformly Elliptic Operators
    Symmetry and non-uniformly elliptic operators Jean Dolbeault ∗ Ceremade, UMR no. 7534 CNRS, Universit´e Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris C´edex 16, France E-mail: [email protected] Internet: http://www.ceremade.dauphine.fr/∼dolbeaul Patricio Felmer ∗; y Departamento de Ingenier´ıa Matem´atica and Centro de Modelamiento Matem´atico, UMR no. 2071 CNRS-UChile Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile E-mail: [email protected] R´egis Monneau ∗ CERMICS, Ecole Nationale des Ponts et Chauss´ees, 6 et 8 avenue Blaise Pascal, Cit´e Descartes, Champs-sur-Marne, 77455 Marne-La-Vall´ee, France E-mail: [email protected] Internet: http://cermics.enpc.fr/∼monneau/home.html Abstract. The goal of this paper is to study the symmetry properties of nonnegative solutions of elliptic equations involving a non uniformly elliptic operator. We consider on a ball the solutions of ∆pu + f(u) = 0 with zero Dirichlet boundary conditions, for p > 2, where ∆p is the p-Laplace operator and f a continuous nonlinearity. The main tools are a comparison result for weak solutions and a local moving plane method which has been previously used in the p = 2 case. We prove local and global symmetry results when u is of class C 1,γ for γ large enough, under some additional technical assumptions. Keywords. Elliptic equations, non uniformly elliptic operators, p-Laplace operator, scalar field equations, monotonicity, symmetry, local symmetry, positivity, non Lipschitz nonlin- earities, comparison techniques, weak solutions, maximum principle, Hopf's lemma, local moving plane method, C 1,α regularity AMS classification (2000).
    [Show full text]