Regular elliptic boundary-value problem in a two-sided refined scale of spaces

TL;DR

This paper investigates a regular elliptic boundary-value problem within a refined scale of Hilbert spaces, establishing its Fredholm property, isomorphisms, a priori estimates, and solution regularity.

Contribution

It introduces the analysis of elliptic boundary-value problems in a two-sided refined scale of Hormander-Volevich-Paneah spaces, extending classical results to this functional framework.

Findings

The problem operator is Fredholm in the refined scale.

The operator generates a complete collection of isomorphisms.

A priori estimates and regularity results for solutions are established.

Abstract

A regular elliptic boundary-value problem over a bounded domain with a smooth boundary is studied. We prove that the operator of this problem is a Fredholm one in the two-sided refined scale of the functional Hilbert spaces and generates a complete collection of isomorphisms. Elements of this scale are the isotropic spaces of Hormander-Volevich-Paneah and some its modifications. A priori estimate for the solution is established and its regularity is investigated.