Riesz bases for $L^2(\partial \Omega)$ and regularity for the Laplace equation in Lipschitz domains

TL;DR

This paper constructs Riesz bases for boundary function spaces in Lipschitz domains to provide an alternative approach to known regularity results for the Laplace equation, enhancing understanding of boundary regularity.

Contribution

It introduces a Hilbertian method to build Riesz bases for boundary spaces, offering a new perspective on classical regularity results for the Laplace equation in Lipschitz domains.

Findings

Constructed two Riesz bases for $L^2(oundary \, \Omega)$

Provided an alternative proof of $H^{1/2}$ regularity results

Enhanced understanding of boundary regularity in Lipschitz domains

Abstract

In a paper from 1996, D. Jerison and C. Kenig among other results provided a $H^{1/2}$ regularity result for the Dirichlet problem for the Laplace equation in Lipschitz domains. In this article, we adopt a Hilbertian approach to construct two Riesz bases for $L^2(\partial \Omega),$ which will allow to find in a different way some of the results of D. Jerison and C. Kenig, and G. Savar\'{e} (1998) about the regularity issue of the Laplace equation.