Boundary Value Problems in the Spaces of Distributions

The present monograph is devoted to the theory of the solvability in generalized functions of general boundary value problems of mathematieal physies. It is the eontinuation of the author's book [Rl], where elliptic boundary value problems have been studied in eomplete seales of spaees of generalized funetions. From the early sixties, in the works of Lions and Magenes [LiM] and Yu. Berezanskii, S. Krein and Va. Roitberg [BKR] the theorems on eomplete eolleetion of isomorphisms have been established. These theorems, roughly speaking, mean that the operator generated by an elliptie boundary value problem establishes an isomorphism between spaees of functions which 'have s derivatives' and spaees functions whieh 'have s - r derivatives' (here s is an arbitrary real number, r is the order of the elliptic problem). The dose results were also obtained by Seheehter [Sehe]. These results and some of their applieations are eontained in the book of Lions and Magenes [LiM2] (see also the survey of Magenes [MagD and Yu. Berezanskii [Ber]. Further progress in the theory under eonsideration was eonneeted, first, with the completion of the dass of elliptie problems for whieh the theorems on eomplete eolleetion of isomorphisms hold, and, henee, with the development of new methods of proving of these theorems, and, second, with the inerease of a number of applieations of the isomorphism theorems. In the author's monograph [RI] the last years' investigations on the isomorphism theorems and some of their applieations have been presented.

Inhalt

0. Introduction.- 1. Green's Formulas and Theorems on Complete Collection of Isomorphisms for General Elliptic Boundary Value Problems for Systems of DouglisNirenberg Structure.- 2. Elliptic Boundary Value Problems for General Systems of Equations with Additional Unknown Functions Defined at the Boundary of a Domain.- 3. Sobolev's Problem.- 4. The Cauchy Problem for General Hyperbolic Systems in the Complete Scale of Sobolev Type Spaces.- 5. Boundary Value and Mixed Problems for General Hyperbolic Systems.- 6. Green's Formula and Density of Solutions for General Parabolic Boundary Value Problems in Functional Spaces on Manifolds.- References.