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The homogeneous and inhomogeneous Dirichlet problem

David S. Jerison, Carlos E. Kenig

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.

The homogeneous and inhomogeneous Dirichlet problem

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.
Paper Structure (5 sections, 24 theorems, 198 equations)

This paper contains 5 sections, 24 theorems, 198 equations.

Table of Contents

  1. Preliminary results from classical harmonic analysis
  2. Analytic and harmonic functions in dimension two
  3. Endpoint counterexample for the inhomogeneous Dirichlet problem
  4. Estimates in all dimensions
  5. Appendix

Key Result

Theorem 1

If $\Omega$ is a bounded, open, convex subset of $\mathbb{R}^n$, then for every $f \in L^2(\Omega)$ the unique function $u \in H_0^1(\Omega)$ such that $\Delta u = f$ satisfies

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Definition 1.1
  • Theorem 1.2: S93
  • Theorem 1.3: S70
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • ...and 36 more