Some semi-homogeneous elliptic boundary-value problems in complete extended Sobolev scale

Abstract

We investigate a regular elliptic boundary-value problem for a homogeneous elliptic equation given in a bounded Euclidean domain with infinitely smooth boundary. We prove that the operator of the problem is bounded and Fredholm in appropriate pairs of H\"ormander inner product spaces. They are parametrized with an arbitrary radial function RO-varying at infifnity in the sense of V.~Avakumovi\'c and form the extended Sobolev scale. We prove that the problem generates a complete collection of isomorphisms on this scale. We also establish a priori estimates for generalized solutions to the problem in these H\"ormander spaces.