Maximal regularity of Dirichlet problem for the Laplacian in Lipschitz domains

Abstract

The focus of this work is on the homogeneous and non-homogeneous Dirichlet problem for the Laplacian in bounded Lipschitz domains (BLD). Although it 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 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 1980s. We identify a functional space E = \{v\in H^{1/2}(\Omega);\nabla v\in [H^1/2(\Omega)]'\} for which the trace operator is continuous from $E$ into $L^2(\Gamma)$. Second, we address the regularity of solutions to the Laplace equation with homogeneous Dirichlet conditions. Using specific equivalent norms in fractional Sobolev spaces and Grisvard's results for polygons and polyhedral domains, we prove that maximal regularity $H^{3/2}$ holds in any BLD $\Omega$, 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 1990s. Third, we describe some criteria which establish new uniqueness results for harmonic functions in Lipschitz domains. In particular, we show that if $u\in H^{1/2}(\Omega)$ or $u\in W^{1, 2N/(N+1)}(\Omega)$, is harmonic in $\Omega$ and vanishes on $\Gamma$, then $u= 0$. These criteria play a central role in deriving regularity properties. Finally, we revisit the classical Area Integral Estimate. Using Grisvard's work and an explicit function given by Necas, we show that this inequality cannot hold in its stated form. Since this estimate 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.