On the smoothness of solutions to elliptic equations in domains with H\"older boundary

TL;DR

This paper investigates how the boundary regularity of domains affects the smoothness of solutions to elliptic boundary value problems, using advanced functional spaces and interpolation methods.

Contribution

It introduces new regularity results for elliptic solutions in domains with H"older boundaries utilizing Sobolev-Slobodetskii and Nikolskii-Besov spaces.

Findings

Established regularity estimates in specialized function spaces.

Extended difference quotient techniques to irregular domains.

Provided a framework for analyzing elliptic solutions in non-smooth domains.

Abstract

The dependence of the smoothness of variational solutions to the first boundary value problems for second order elliptic operators are studied. The results use Sobolev-Slobodetskii and Nikolskii-Besov spaces and their properties. Methods are based on real interpolation technique and generalization of Savar\'{e}-Nirenberg difference quotient technique.