Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces

TL;DR

This paper studies elliptic boundary-value problems with boundary unknowns on refined function spaces, establishing boundedness, Fredholm properties, and regularity results that lead to new conditions for classical differentiability of solutions.

Contribution

It introduces analysis of Lawruk's elliptic problems on two-sided refined scales of Hörmander spaces, proving boundedness, Fredholmness, and regularity results in these advanced function spaces.

Findings

Operator is bounded and Fredholm on refined scales.

Solutions exhibit local regularity and a priori estimates.

New conditions for solutions to have classical derivatives.

Abstract

We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on two-sided refined scales built on the base of the isotropic H\"ormander inner product spaces. The regularity of the distributions forming these spaces are characterized by a real number and an arbitrary function that varies slowly at infinity in the sense of Karamata. For the generalized solutions to the problem, we prove theorems on a priori estimates and local regularity in these scales. As applications, we find new sufficient conditions under which the solutions have continuous classical derivatives of a prescribed order.