The Dirichlet Problem for the Laplacian in Lipschitz Domains Revisited Dedicated to the memory of my friend Mohand Moussaoui

Overview & Synopsis

The purpose of this work is to study the Dirichlet problem:-∆u = f in Ω and u = g on Γ,with data belonging to appropriate Sobolev spaces, when the domain Ω in R N , with N ≥ 2, is assumed to be only Lipschitz. Although this has been extensively studied by many authors, we would like to return to a number of fundamental questions and known results, such as the traces, the uniqueness and the maximal regularity of solutions. First, to treat non-homogeneous boundary conditions, we rigorously define the notion of traces for non regular functions. This approach replaces the non-tangential trace notion that has dominated the literature since the 80s. We identify a functional spacewhich satisfies the embeddings H 1/2 00 (Ω) ↩→ E(∇; Ω) ↩→ H 1/2 (Ω) and the trace operator γ : E(∇; Ω) → L 2 (Γ) is well defined, continuous and leads to a new characterization of H 1/2 00 (Ω) as precisely the kernel of this operator. Second, by using Grisvard's results and interpolation theory, we prove the following inequality: let Ω be a polygonal domain, with K angles ω 1 , . . . , ω K larger than π,where for s = 1/2 the space H -1/2 (Ω) denotes the dual space of H 1/2 00 (Ω) and C P,L (Ω) depends only on s, on Poincaré and Lipschitz constants of Ω. The same estimate holds for polyhedral domains with dihedral angle openings (toward the interior of Ω) larger than π. By virtue of the inequality (0.1), the above characterization of H 1/2 00 (Ω) and the uniqueness of H 1/2 (Ω) solution to Problem (L D ), we prove that maximal regularity H 3/2 holds, in any bounded Lipschitz domain Ω, for all right-hand sides in the dual of H 1/2 00 (Ω). This conclusion contradicts the prevailing claims in the