Lawruk elliptic boundary-value problems for homogeneous differential equations

Abstract

We investigate Lawruk elliptic boundary-value problems for homogeneous differential equations in a two-sided refined Sobolev scale. These problems contain additional unknown functions in the boundary conditions of arbitrary orders. The scale consists of inner-product H\"ormander spaces whose orders of regularity are given by any real number and a function which varies slowly at infinity in the sense of Karamata. We establish theorems on the Fredholm property of the problems in the refined Sobolev scale and on local regularity and local a priori estimate (up to the boundary of the domain) of their generalized solutions. We find sufficient conditions under which components of these solutions are $l\geq0$ times continuously differentiable functions.