The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains

Abstract

Given a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, we prove that the Poisson's problem for the Laplacian with right-hand side in $L_{-t}^p(\Omega )$, Robin-type boundary datum in the Besov space $B_p^{1-1/p-t,p}(\partial \Omega )$ and non-negative, non-everywhere vanishing Robin coefficient $b\in L^{n-1}(\partial \Omega )$, is uniquely solvable in the class $L_{2-t}^p(\Omega )$ for $(t,\frac{1}{p})\in {\mathcal{V}}_{ϵ}$, where ${\mathcal{V}}_{ϵ}$ ($ϵ\ge 0$) is an open ($\Omega$,$b$)-dependent plane region and ${\mathcal{V}}_0$ is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson's problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.