The Cauchy problem for solutions of elliptic equations.

*(English)*Zbl 0831.35001
Mathematical Topics. 7. Berlin: Akademie-Verlag. 478 p. (1995).

This book undertakes the first steps towards a general theory of the Cauchy problem for solutions of elliptic equations of high order as well as solutions of overdetermined systems of differential equations with injective symbols. The study of the Cauchy problem is carried out in three directions: 1) determining the “degree of instability”, which is connected with sharp theorems on approximation by solutions of systems with surjective symbols; 2) findinig solvability conditions, which is based on development of Hilbert space methods in the Cauchy problem; and 3) constructing approximate solutions, which requires elaborating efficient ways of approximation by solutions of systems with surjective symbols. This book consist of an introduction and 12 chapters:

Elements of the theory of distributions and the function spaces theory are considered in Chapter 1. Chapter 2 is devoted to the theory of pseudodifferential operators in spaces of distributions on compact sets and, in the main, the boundedness theorems. In Chapter 3 the concept of a capacity, associated with a seminormed space of distributions, is explained. Chapter 4 is devoted to a generalization of the algebraic condition of ellipticity to the context of overdetermined systems. In Chapters 5-8 the problem of approximation on compacta by solutions of systems with surjective symbols is considered. In Chapter 9 generalized boundary values of solutions of systems with injective symbols are studied. In Chapter 10 the Cauchy problem is investigated in the class of distributions of finite order of growth near the boundary. In Chapter 11 the classical method of Fisher-Riesz equations is developed to derive both a solvability condition of the Cauchy problem and an approximate formula for solutions. The final chapter presents another Hilbert space approach to the Cauchy problem for solutions of a system with injective symbol. Here so-called “bases with double orthogonality” are used.

A large part of the material appears for the first time in book form. This book will be useful both for scientists and post graduat students, and for higher courses students. It may be also used for special courses in universities.

Elements of the theory of distributions and the function spaces theory are considered in Chapter 1. Chapter 2 is devoted to the theory of pseudodifferential operators in spaces of distributions on compact sets and, in the main, the boundedness theorems. In Chapter 3 the concept of a capacity, associated with a seminormed space of distributions, is explained. Chapter 4 is devoted to a generalization of the algebraic condition of ellipticity to the context of overdetermined systems. In Chapters 5-8 the problem of approximation on compacta by solutions of systems with surjective symbols is considered. In Chapter 9 generalized boundary values of solutions of systems with injective symbols are studied. In Chapter 10 the Cauchy problem is investigated in the class of distributions of finite order of growth near the boundary. In Chapter 11 the classical method of Fisher-Riesz equations is developed to derive both a solvability condition of the Cauchy problem and an approximate formula for solutions. The final chapter presents another Hilbert space approach to the Cauchy problem for solutions of a system with injective symbol. Here so-called “bases with double orthogonality” are used.

A large part of the material appears for the first time in book form. This book will be useful both for scientists and post graduat students, and for higher courses students. It may be also used for special courses in universities.

Reviewer: Ya.A.Rojtberg (Chernigov)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J45 | Systems of elliptic equations, general (MSC2000) |

32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |