The aim of this book is to give a systematic study of questions con­ cerning existence, uniqueness and regularity of solutions of linear partial differential equations and boundary problems. Let us note explicitly that this program does not contain such topics as eigenfunction expan­ sions, although we do give the main facts concerning differential operators which are required for their study. The restriction to linear equations also means that the trouble of achieving minimal assumptions concerning the smoothness of the coefficients of the differential equations studied would not be worth while; we usually assume that they are infinitely differenti­ able. Functional analysis and distribution theory form the framework for the theory developed here. However, only classical results of functional analysis are used. The terminology employed is that of BOURBAKI. To make the exposition self-contained we present in Chapter I the elements of distribution theory that are required. With the possible exception of section 1.8, this introductory chapter should be bypassed by a reader who is already familiar with distribution theory.

Inhalt

I: Functional analysis.- I. Distribution theory.- 1.0. Introduction.- 1.1. Weak derivatives.- 1.2. Test functions.- 1.3. Definitions and basic properties of distributions.- 1.4. Differentiation of distributions and multiplication by functions.- 1.5. Distributions with compact support.- 1.6. Convolution of distributions.- 1.7. Fourier transforms of distributions.- 1.8. Distributions on a manifold.- II. Some special spaces of distributions.- 2.0. Introduction.- 2.1. Temperate weight functions.- 2.2. The spaces ?p, k.- 2.3. The spaces $$\mathcal{B}{p,k}^{loc}$$.- 2.4. The spaces ?(s).- 2.5. The spaces ?(m, s).- 2.6. The spaces $$\mathcal{H}{(s)}^{loc}\left( \Omega \right)$$ when ? is a manifold.- II: Differential operators with constant coefficients.- III. Existence and approximation of solutions of differential equations.- 3.0. Introduction.- 3.1. Existence of fundamental solutions.- 3.2. The equation P (D) u = f when f ? ??.- 3.3. Comparison of differential operators.- 3.4. Approximation of solutions of homogeneous differential equations.- 3.5. The equation P (D) u = f when f is in a local space $$ \subset {\mathcal{D}'_F}$$.- 3.6. The equation P (D) u = f when $$f \in \mathcal{D}'$$.- 3.7. The geometric meaning of P-convexity and strong P-convexity.- 3.8. Systems of differential operators.- IV. Interior regularity of solutions of differential equations.- 4.0. Introduction.- 4.1. Hypoelliptic operators.- 4.2. Partially hypoelliptic operators.- 4.3. Partial hypoellipticity at the boundary.- 4.4. Estimates for derivatives of high order.- V. The Cauchy problem (constant coefficients).- 5.0. Introduction.- 5.1. The classical existence theory for analytic data.- 5.2. The non-uniqueness of the characteristic Cauchy problem.- 5.3. Holmgren's uniqueness theorem.- 5.4. The necessity of hyperbolicity for the existence of solutions to the noncharacteristic Cauchy problem.- 5.5. Algebraic properties of hyperbolic polynomials.- 5.6. The Cauchy problem for a hyperbolic equation.- 5.7. A global uniqueness theorem.- 5.8. The characteristic Cauchy problem.- III: Differential operators with variable coefficients.- VI. Differential equations which have no solutions.- 6.0. Introduction.- 6.1. Conditions for non-existence.- 6.2. Some properties of the range.- VII. Differential operators of constant strength.- 7.0. Introduction.- 7.1. Definitions and basic properties.- 7.2. Existence theorems when the coefficients are merely continuous.- 7 3 Existence theorems when the coefficients are in C?.- 7.4. Hypoellipticity.- 7.5. The analyticity of the solutions of elliptic equations.- VIII. Differential operators with simple characteristics.- 8.0. Introduction.- 8.1. Necessary conditions for the main estimates.- 8.2. Differential quadratic forms.- 8.3. Estimates for elliptic operators.- 8.4. Estimates for operators with real coefficients.- 8.5. Estimates for principally normal operators.- 8.6. Pseudo-convexity.- 8.7. Estimates, existence and approximation theorems in ?(s).- 8.8. The unique continuation of singularities.- 8.9. The uniqueness of the Cauchy problem.- IX. The Cauchy problem (variable coefficients).- 9.0. Introduction.- 9.1. Preliminary lemmas.- 9.2. The basic L2 estimate.- 9.3. Existence theory for the Cauchy problem.- X. Elliptic boundary problems.- 10.0. Introduction.- 10.1. Definition of elliptic boundary problems.- 10.2. Preliminaries concerning ordinary differential operators.- 10.3. Construction of a parametrix.- 10.4. Local theory of elliptic boundary problems.- 10.5. Elliptic boundary problems in a compact manifold with boundary.- 10.6. Various extensions and remarks.- Appendix. Some algebraic lemmas.- Index of notations.