Elliptic Operator Theory
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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 >> 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. -
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. -
Hodge Decomposition
Hodge Decomposition Daniel Lowengrub April 27, 2014 1 Introduction All of the content in these notes in contained in the book Differential Analysis on Complex Man- ifolds by Raymond Wells. The primary objective here is to highlight the steps needed to prove the Hodge decomposition theorems for real and complex manifolds, in addition to providing intuition as to how everything fits together. 1.1 The Decomposition Theorem On a given complex manifold X, there are two natural cohomologies to consider. One is the de Rham Cohomology which can be defined on a general, possibly non complex, manifold. The second one is the Dolbeault cohomology which uses the complex structure. We’ll quickly go over the definitions of these cohomologies in order to set notation but for a precise discussion I recommend Huybrechts or Wells. If X is a manifold, we can define the de Rahm Complex to be the chain complex 0 E(Ω0)(X) −d E(Ω1)(X) −d ::: Where Ω = T∗ denotes the cotangent vector bundle and Ωk is the alternating product Ωk = ΛkΩ. In general, we’ll use the notation! E(E) to! denote the sheaf! of sections associated to a vector bundle E. The boundary operator is the usual differentiation operator. The de Rham cohomology of the manifold X is defined to be the cohomology of the de Rham chain complex: d Ker(E(Ωn)(X) − E(Ωn+1)(X)) n (X R) = HdR , d Im(E(Ωn-1)(X) − E(Ωn)(X)) ! We can also consider the de Rham cohomology with complex coefficients by tensoring the de Rham complex with C in order to obtain the de Rham complex! with complex coefficients 0 d 1 d 0 E(ΩC)(X) − E(ΩC)(X) − ::: where ΩC = Ω ⊗ C and the differential d is linearly extended. -
An Introduction to Pseudo-Differential Operators
An introduction to pseudo-differential operators Jean-Marc Bouclet1 Universit´ede Toulouse 3 Institut de Math´ematiquesde Toulouse [email protected] 2 Contents 1 Background on analysis on manifolds 7 2 The Weyl law: statement of the problem 13 3 Pseudodifferential calculus 19 3.1 The Fourier transform . 19 3.2 Definition of pseudo-differential operators . 21 3.3 Symbolic calculus . 24 3.4 Proofs . 27 4 Some tools of spectral theory 41 4.1 Hilbert-Schmidt operators . 41 4.2 Trace class operators . 44 4.3 Functional calculus via the Helffer-Sj¨ostrandformula . 50 5 L2 bounds for pseudo-differential operators 55 5.1 L2 estimates . 55 5.2 Hilbert-Schmidt estimates . 60 5.3 Trace class estimates . 61 6 Elliptic parametrix and applications 65 n 6.1 Parametrix on R ................................ 65 6.2 Localization of the parametrix . 71 7 Proof of the Weyl law 75 7.1 The resolvent of the Laplacian on a compact manifold . 75 7.2 Diagonalization of ∆g .............................. 78 7.3 Proof of the Weyl law . 81 A Proof of the Peetre Theorem 85 3 4 CONTENTS Introduction The spirit of these notes is to use the famous Weyl law (on the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold) as a case study to introduce and illustrate one of the many applications of the pseudo-differential calculus. The material presented here corresponds to a 24 hours course taught in Toulouse in 2012 and 2013. We introduce all tools required to give a complete proof of the Weyl law, mainly the semiclassical pseudo-differential calculus, and then of course prove it! The price to pay is that we avoid presenting many classical concepts or results which are not necessary for our purpose (such as Borel summations, principal symbols, invariance by diffeomorphism or the G˚ardinginequality). -
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. -
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. -
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. -
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. -
The Heat Kernel Dexter Chua
The Heat Kernel Dexter Chua 1 The Heat Equation 3 2 Heat Kernel for the Laplacian 8 3 Heat Kernel on an Unbounded Domain 13 4 The Hirzebruch Signature Theorem 17 5 Signature for Manifolds with Boundary 19 6 Heat Equation on the Boundary 22 7 Heat Equation on Manifold with Boundary 27 Appendix A Index and Geometry 30 On a closed Riemannian manifold, there are three closely related objects we can study: (i) Topological invariants such as the Euler characteristic and the signature; (ii) Indexes of differential operators; and (iii) Differential forms on the manifold. (i) and (ii) are mainly related by the Hodge decomposition theorem | the signature and Euler characteristic count the dimensions of the cohomology groups, and Hodge theory says the cohomology groups are exactly the kernels of the Laplacian. This is explored briefly in Appendix A. (i) and (iii) are related to each other via results such as the Gauss{Bonnet theorem and the Hirzebruch signature theorem. The Gauss{Bonnet theorem says the Euler characteristic of a surface is the integral of the curvature; The Hirzebruch signature theorem says the signature is the integral of certain differential forms called the L-genus, given as a polynomial function of the Pontryagin forms. (ii) and (iii) can also be related directly to each other, and this connection is what I would call \index theory". The main theorem is the Atiyah{Singer index theorem, 1 and once we have accepted the connection between (i) and (ii), we can regard the Gauss{Bonnet theorem and the Hirzebruch signature theorem as our prototypical examples of index theory. -
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. -
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. -
The Atiyah-Singer Index Formula
Part 3 The Atiyah-Singer Index Formula “Inperhapsmostcaseswhenwefailto findtheanswertoaquestion,thefailure iscausedbyunsolvedorinsufficiently solvedsimplerandeasierproblems.Thus alldependsonfindingtheeasierproblem andsolvingitwithtoolsthatareasperfect aspossibleandwithnotionsthatarecapable ofgeneralization.” (D.Hilbert,1900) CHAPTER 14 Introduction to Algebraic Topology (K-Theory) It is the goal of this part to develop a larger portion of algebraic topology by means of a theorem of Raoul Bott concerning the topology of the general linear group GL(N,C) on the basis of linear algebra, rather than the theory of “simplicial complexes” and their “homology” and “cohomology”. There are several reasons for doing so. First of all, is of course a matter of taste and familiarity as to which ap- proach “codifying qualitative information in algebraic form” (Atiyah) one prefers. In addition, there are objective criteria such as simplicity, accessibility and trans- parency, which speak for this path to algebraic topology. Finally, it turns out that this part of topology is most relevant for the investigation of the index problem. Before developing the necessary machinery, it seems advisable to explain some basic facts on winding numbers and the topology of the general linear group GL(N,C). Note that the group GL(N,C) moved to fore in Part 2 already in connection with the symbol of an elliptic operator, and that the group Z of integers was in a certain sense the topic of Part 1, the Fredholm theory. In the following, Part 3, the concern is (roughly) the deeper connection between the previous parts. Thereby we will be guided by the search for the “correct” and “promising” generalizations of the theo- rem of Israil Gohberg and Mark Krein on the index of Wiener-Hopf operators.