The Dirichlet Problem for the Laplacian in Lipschitz Domains Revisited

Abstract

Here we study the Dirichlet problem for the Laplacian, we denote $(\mathscr{L}_D)$, when the domain $\Omega$ in $\mathbb{R}^N,$ with $N \geq 2$, 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(\nabla;\, \Omega)$ which satisfies the embeddings $H^{1/2}_{00}(\Omega)\hookrightarrow E \hookrightarrow H^{1/2}(\Omega)$ and the trace operator $\gamma: E\rightarrow L^2(\Gamma)$ is well defined, continuous and leads to a new characterization of $H^{1/2}_{00}(\Omega)$. Second, by using Grisvard's results, interpolation theory, the characterization of $H^{1/2}_{00}(\Omega)$ and the uniqueness of $H^{1/2}(\Omega)$ solution to Problem $(\mathscr{L}_D)$, we prove that maximal regularity $H^{3/2}$ holds for all right-hand sides in the dual of $H^{1/2}_{00}(\Omega)$. 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 $W^{s, p}(\Omega)$ to the problem $(\mathscr{L}_D)$. Finally, we revisit the classical Area Integral Estimate of Dahlberg for a harmonic function $u$ in $\Omega$ vanishing at some interior point: \begin{equation}\label{ineg} \int_\Gamma \vert u \vert^2 d\sigma \leq C \int_\Gamma \vert S(u)\vert^2 d\sigma \simeq C \int_\Omega \varrho \vert\nabla u\vert^2 dx. \end{equation} We show that this inequality cannot hold in its stated form. Since the estimate \eqref{ineg} has been widely used to argue that $H^{3/2}$-regularity is unattainable for data in the dual of $H^{1/2}_{00}(\Omega)$, our counterexample provides a decisive clarification.