Linear Differential Operators with Constant Coefficients

ISBN-13: 9783642462214 / Angielski / Miękka / 2012 / 448 str.

Victor Pavlovic Palamodov;Arthur A. Brown

Opis

This book contains a systematic exposition of the facts relating to partial differential equations with constant coefficients. The study of systems of equations in general form occupies a central place. Together with the classical problems of the existence, the uniqueness, and the regularity of the solutions, we also consider the specific problems that arise in connection with overdetermined and underdetermined systems of equations: the extendabiIity of the solutions into a wider region, the extendability of regularity, M-cohomology and so on. Great attention is paid to the connections and the parallels with the theory of functions of several complex variables. The choice of material was dictated by a number of considerations. Among all the facts relating to general systems of equations, the book contains none that relate to the behavior of differential operators in spaces of slowly growing functions. Missing also are results relating to a single equation in one unknown function: the correctness of the Cauchy problem, certain theorems on p-convexity, and the theory of boundary values, are all set forth in other monographs (Gel'fand and Silov 3], Hormander 10] and Treves 4]). The book consists of two parts. In the first, we set forth the analytic method which forms the basis for the contents of the second part, which itself is dedicated to differential equations. The first part is pre- ceded by an introduction in which the content and methods of Part I are described. All the notes and bibliographical references are collected together in a special section.

Spis treści

§1. Exponential representation for an ordinary equation with one unknown function.- §2. Exponential representation of the solutions of partial differential equations.- §3. The exponential representation of solutions of arbitrary systems.- I. Analytic Methods.- I. Homological Tools.- §1. Families of topological modules.- §2. The fundamental homology theorem.- §3. Operations on modules.- II. Division with Remainder in the Space of Power Series.- §1. The space of power series.- §2. The base sequence of matrices.- §3. Stabilization of the base sequence.- §4. p-decompositions.- III. Cohomologies of Analytic Functions of Bounded Growth.- §1. The space of holomorphic functions.- §2. The operator $${D_{\bar z}}$$ in spaces of type I.- §3. M-cohomologies.- §4. The theorem on the triviality of M-cohomologies.- §5. Cohomologies connected with P-matrices.- IV. The Fundamental Theorem.- §1. Some properties of finite P-modules.- §2. Local p-operators.- §3. The fundamental inequality for the operator D.- §4. Noetherian operators.- §5. The fundamental theorem.- II. Differential Equations with Constant Coefficients.- V. Linear Spaces and Distributions.- §1. Limiting processes in families of linear spaces.- §2. Functional spaces.- §3. Fourier transforms.- VI. Homogeneous Systems of Equations.- §4. The exponential representation of solutions of homogeneous sytems of equations.- § 5. Hypoelliptic operators.- §6. Uniqueness of solutions of the Cauchy problem.- VII. Inhomogeneous Systems.- §7. Solubility of inhomogeneous systems. M-convexity..- §8. M-convexity in convex regions.- §9. The connection between M-convexity and the properties of a sheaf of solutions of a homogeneous system.- § 10. The algebraic conditions for M-convexity.- §11. Geometrical conditions of M-convexity.- §12. Operators of the form $$p\left( {{D_{\bar \xi }}} \right)$$ in domains of holomorphy.- VIII. Overdetermined Systems.- § 13. Concerning the modules Exti (M, P).- § 14. The extension of solutions of homogeneous systems.- § 15. The influence of boundary values on the behavior of the solutions within a region.- Notes.- Index of Basic Notation.