Elliptic problems with unknowns on the boundary and irregular boundary data

TL;DR

This paper studies elliptic boundary problems with unknown boundary conditions and irregular data, analyzing local regularity of solutions within refined Sobolev spaces and establishing new theorems on solution properties.

Contribution

It introduces new regularity theorems for elliptic problems with unknown boundary conditions using refined Sobolev scales, extending classical Sobolev space results.

Findings

Proved local regularity theorems for solutions.

Established local a priori estimates.

Extended results to refined Sobolev spaces.

Abstract

We consider an elliptic problem with unknowns on the boundary of the domain of the elliptic equation and suppose that the right-hand side of this equation is square integrable and that the boundary data are arbitrary (specifically, irregular) distributions. We investigate local (up to the boundary) properties of generalized solutions to the problem in Hilbert distribution spaces that belong to the refined Sobolev scale. These spaces are parametrized with a real number and a function that varies slowly at infinity. The function parameter refines the number order of the space. We prove theorems on local regularity and a local a priori estimate of generalized solutions to the problem under investigation. These theorems are new for Sobolev spaces as well.