Elliptic problems in the sense of Lawruk with boundary operators of higher orders in refined Sobolev scale

Abstract

In a refined Sobolev scale, we investigate an elliptic boundary-value problem with additional unknown functions in boundary conditions for which the maximum of orders of boundary operators is grater than or equal to the order of the elliptic equation. This scale consists of inner product Hörmander spaces whose order of regularity is given by a real number and a function varying slowly at infinity in the sense of Karamata. We prove a theorem on the Fredholm property of a bounded operator corresponding to this problem in the refined Sobolev scale. For the generalized solutions to the problem, we establish a local a priory estimate and prove a theorem about their regularity in Hörmander spaces. We find sufficient conditions under which given generalized derivatives of the solutions are continuous.