Elliptic B. Lawruk boundary value problems in the extended Sobolev scale

Abstract

In the extended Sobolev scale, we investigate a class of elliptic problems with additional unknown functions in boundary conditions, which is introduced by B.~Lawruk. This scale consists of all Hilbert spaces that are interpolation spaces for pairs of Sobolev inner-product spaces and, moreover, admits a constructive description in terms of H\"ormander spaces. We prove a theorem on the Fredholm property of the bounded operators corresponding to these problems on the extended Sobolev scale and a theorem on local regularity of their solutions in H\"ormander spaces. We find sufficient conditions under which the generalized derivatives (of a given order) of the solutions are continuous.