Elliptic Boundary Value Problems in the Spaces of Distributions: 384 (Mathematics and Its Applications, 384) - Hardcover

Roitberg, Y.

Hardcover

ISBN 10: 0792343034 ISBN 13: 9780792343035

Publisher: Springer, 1996

Synopsis

(2) Green's functions were constructed and studied for general elliptic boundary-va­ lue problems ([BeRI]-[Ber3], [Kovl], and [Kov2]). (3) Generalized eigenfunctions, spectral function, etc., were investigated in spectral theory [Ber]. (4) Elliptic boundary-value problems with arbitrary power singularities on the right­ hand sides were investigated ([Roi6], [Kos2,] and [Ser)). (5) A family of elliptic problems was studied in a family of expanding domains; the obtained results were applied to the investigation of elliptic problems with time deriva­ tives in the boundary conditions and parabolic problems in noncylindrical domains ([Kre3], [KLa], and [IKK)). (6) Strongly degenerate elliptic problems were studied in [RShI5]-[RShI8]. (7) Application to the investigation of traces of generalized solutions of equations on the boundary of the domain ([Roi8], [Roi4], [Roil5], and [RoiI7D. (8) Application to problems of transmission ([Roi3], [Roi4], [RSh6], and [RShlOD. (9) Applications to the investigation of nonlocal elliptic problems ([RSh7]-[RSh9], [RShll], and [RShI2)). (10) Application to problems of optimal control ([Li05], [RSh13], and [RShI4D. (11) Application to the theory of underdetermined and overdetermined problems ([Klvl]-[KLv3D. (12) Application to nonlinear problems ([Lio6] and [KrSD. (13) Application to the problems of mechanical motion of non-Newton liquids [Lit]. These and many other examples demonstrate that the efforts of the reader who wants to become an expert in the theory developed in the book will give fruitful results very quickly.

About this title

Review

`The book is well written and deals well with a topic which requires great care and attention to detail. It is an excellent digest of known results for this problem and a valuable reference for those who wish to apply known results for problem (1) to various other disciplines.'
SIAM Review, 40:1 (1998)

Synopsis

This volume endeavours to summarise all available data on the theorems on isomorphisms and their ever increasing number of possible applications. It deals with the theory of solvability in generalised functions of general boundary-value problems for elliptic equations. In the early sixties, Lions and Magenes, and Berezansky, Krein and Roitberg established the theorems on complete collection of isomorphisms. Further progress of the theory was connected with proving the theorem on complete collection of isomorphisms for new classes of problems, and hence with the development of new methods to prove these theorems. The theorems on isomorphisms were first established for elliptic equations with normal boundary conditions. However, after the Noetherian property of elliptic problems was proved without assuming the normality of the boundary expressions, this became the natural way to consider the problems of establishing the theorems on isomorphisms for general elliptic problems.The present author's method of solving this problem enabled proof of the theorem on complete collection of isomorphisms for the operators generated by elliptic boundary-value problems for general systems of equations.

This monograph will be of interest to mathematicians whose work involves partial differential equations, functional analysis, operator theory and the mathematics of mechanics.

Edition details

• Publisher: Springer
• Publication date: 1996
• Language: English
• ISBN 10: 0792343034
• ISBN 13: 9780792343035
• Binding: Hardcover
• Number of pages: 431