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

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 Hörmander spaces Hs,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 Hs,φ (Ω), s < ord A, and f ϵ L2(Ω). We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.