On Approximation by Solutions of non-local Elliptic Problems

Abstract

In this paper we study the question whether an arbitrary function on a manifold may be approximated by solutions of a non-local boundary value problem for an elliptic equation of order 2m with discontinuous coefficients (the precise statement of the question is given in Section 3). Such questions for usual elliptic boundary value problems were studied by many authors since 1960 (see [1], [2], [3], [10], [11] and their references). The mentioned non-local elliptic problems are studied since 1970 when they were introduced by Ya. Roitberg and the author in [4]; similar problems arise for instance in magnetohydrodynamics. The definition of such problems is given in Section 2.