The homogeneous and inhomogeneous Dirichlet problem

TL;DR

This paper clarifies misconceptions about the Dirichlet problem for the Laplacian on Lipschitz domains, correcting recent claims and reaffirming established estimates through multiple proofs.

Contribution

It identifies a critical flaw in recent contradictory claims and provides a self-contained proof of Dahlberg's results, reaffirming classical estimates for the Dirichlet problem.

Findings

Disproof of recent contradictory claims

Validation of Dahlberg's area integral estimates

Multiple independent proofs of key results

Abstract

We revisit the homogeneous and inhomogeneous Dirichlet problem for the Laplacian on Lipschitz domains. This is motivated by the recent postings by Amrouche and Moussaoui which purport to contradict known area integral estimates of Dahlberg and known Sobolev space estimates of Jerison and Kenig. We explain the fatal gap in the reasoning in these postings and give a self-contained proof of a special case of the results of Dahlberg. We then show that this is sufficient to disprove the central conclusions of these postings. We also provide two further proofs of the results of Dahlberg, adapted from work of Kenig and of Dahlberg-Kenig-Pipher-Verchota. Other proofs are in the original paper by Dahlberg and in work by Fabes-Mendez-Mitrea.