The Dirichlet problem for the Laplacian in Lipschitz domain. Abstract

TL;DR

This paper investigates boundary value problems for the Laplacian and bi-Laplacian in Lipschitz domains within Sobolev spaces, aiming to improve existing theories and achieve maximal regularity with simpler techniques.

Contribution

It provides new insights and improvements on boundary value problems in Lipschitz domains, especially for critical Sobolev space indices, using less complex methods.

Findings

Enhanced regularity results for boundary value problems

Improved understanding of Sobolev space behavior on Lipschitz domains

Achieved maximal regularity and optimality in solutions

Abstract

The main purpose of this paper is to address some questions concerning boundary value problems related to the Laplacian and bi-Laplacian operators, set in the framework of classical $H^s$ Sobolev spaces on a bounded Lipschitz domain of R^N. These questions are not new and a lot of work has been done in this direction by many authors using various techniques since the 80's. If for regular domains almost every thing is elucidated, it is not the case for Lipschitz ones and for $s$ of the form $s = k + 1/2$, with $k$ integer. It is well known that this framework is delicate. Even in these cases many results are well established but sometimes not satisfactory. Several questions remain posed. Our main goal through this work is on one hand to give some improvements to the theory and on another one by using techniques which do not require too intricate calculations. We also tried to obtain maximal regularity for the solutions and as far as we can optimality of the results.