2025/09/10 by Chérif Amrouche, Mohand Moussaoui, Amrouche, Chérif +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2509.08543
Here we study the Dirichlet problem for the Laplacian, we denote (\mathscrLD), when the domain Ω in ℝN, with N ≥ 2, is assumed to be only Lipschitz. 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, we rigorously define the notion of traces for non regular functions. This approach replaces the non-tangential trace notion. We identify a functional space E(∇; Ω) which satisfies the embeddings H1/200(Ω)\hookrightarrow E \hookrightarrow H1/2(Ω) and the trace operator γ: E→ L2(Γ) is well defined, continuous and leads to a new characterization of H1/200(Ω). Second, by using Grisvard's results, interpolation theory, the characterization of H1/200(Ω) and the uniqueness of H1/2(Ω) solution to Problem (\mathscrLD), we prove that maximal regularity H3/2 holds for all right-hand sides in the dual of H1/200(Ω). This conclusion contradicts the prevailing claims in the literature since the 90s. Third, we return to the very delicate question of the existence and uniqueness of solutions Ws, p(Ω) to the problem (\mathscrLD). Finally, we revisit the classical Area Integral Estimate of Dahlberg for a harmonic function u in Ω vanishing at some interior point: ∫Γ\vert u \vert2 dσ≤ C ∫Γ\vert S(u)\vert2 dσ≃ C ∫Ω\varrho \vert∇ u\vert2 dx. We show that this inequality cannot hold in its stated form. Since the estimate \eqrefineg has been widely used to argue that H3/2-regularity is unattainable for data in the dual of H1/200(Ω), our counterexample provides a decisive clarification.