Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces

Abstract

We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hormander spaces H s,φ, which form the refined Sobolev scale. The order of differentiation for these spaces is given by a real number s and a positive function φ slowly varying at infinity in Karamata’s sense. We consider this problem for an arbitrary elliptic equation Au = f in a bounded Euclidean domain Ω under the condition that u ϵ H s,φ (Ω), s < ord A, and f ϵ L 2 (Ω). We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.