*(English)*Zbl 0946.35113

From the introduction: This book is an introduction to the theory of boundary value problems for elliptic functional differential equations. It is mainly based on the investigations of the author and on the courses he gave to students at the Moscow State Aviation Institute. It is addressed to graduate students and mathematicians with interests in functional differential equations and partial differential equations. It could also be useful to specialists in the fields of control theory, theory of diffusion processes, and elasticity theory. We consider three types of differential-difference operator: 1) strongly elliptic operators (Chapters II, III), 2) symmetric semi-bounded operators with degeneration (Chapter IV), and 3) elliptic operators (Chapter V).

In cases 1) and 2), we study the following basic questions: properties of corresponding sesquilinear forms, solvability and spectrum, and smoothness of generalized solutions. The Friedrichs extension of sectorial operators and the theory of difference operators provide a most convenient tool for the study of these problems.

In case 3), we reduce a boundary value problem for elliptic differential-difference equation to an elliptic differential equation with nonlocal conditions. Therefore, we study a priori estimates, solvability and the spectrum for elliptic differential equations with nonlocal boundary conditions. Combining these results and properties of difference operators, we consider solvability, spectrum and smoothness of generalized solutions of boundary value problems for elliptic differential-difference equations.

The most interesting properties of elliptic differential-difference equations are connected with smoothness of generalized solutions. Unlike elliptic differential equations, the smoothness of generalized solutions of elliptic differential-difference equations can be violated in a bounded domain $Q\subset {\mathbb{R}}^{n}$, even for infinitely differentiable right-hand sides. In general, the smoothness of solutions is conserved only in certain subdomains. The presentation of these new results is one of the main purposes of this book.

The book consists of five chapters. I) Boundary value problems for functional differential equations in one dimension. II) The first boundary value problem for strongly elliptic differential difference equations. III) Applications to the mechanics of a deformable body. IV) Semi-bounded differential-difference operators with degeneration. V) Nonlocal elliptic boundary value problems.

The Appendices A–C are devoted to the elements of functional analysis, the theory of Sobolev spaces, and the theory of elliptic differential equations, respectively. The book contains a list of symbols and an index.