Solvability and Regularity of Solutions for Some Classes of Elliptic Functional-Differential Equations

Abstract

1. This review is devoted to the theory of boundary-value problems for some classes of elliptic functionaldifferential equations. Boundary-value problems for elliptic functional-differential equations have some astonishing properties. For example, unlike elliptic differential equations, the smoothness of generalized solutions can be broken in a bounded domain and is preserved only in some subdomains. The symbol of a self-adjoint semibounded functional-differential operator can change its sign. Boundary-value problems for elliptic functional-differential equations are closely related to nonlocal elliptic problems. On the other hand, elliptic functional-differential equations have important applications to nonlinear optics, elasticity theory, control theory, and diffusion processes. Elliptic functional-differential equations containing transformations of arguments were studied by A. B. Antonevich [2], D. Przeworska-Rolewicz [36], and V. S. Rabinovich [37]. The authors assumed that the transformations of arguments map a domain onto itself. Therefore, their results were similar to the well-known results for elliptic differential equations. The situation changes if the equation has these shifts in the highest derivatives, and the shifts map the points of the boundary into the domain. For an introduction to the theory of these equations, see A. L. Skubachevskii [67]. 2. In this paper, we mainly study boundary-value problems for elliptic differential-difference equations and for elliptic functional-differential equations with contractions and expansions. In order to illustrate the properties of elliptic differential-difference equations, we consider the following example: