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

Abstract

Given a bounded Lipschitz domain $\Omega\subset {\mathbb R}^n$, $n\geq 3$, we prove~that the Poisson's problem for the Laplacian with right-hand side in $L^p_{-t}(\Omega)$, Robin-type boundary datum in the Besov space $B^{1-1/p-t,p}_{p}(\partial \Omega)$ and non-negative, non-everywhere vanishing Robin coefficient $b\in L^{n-1}(\partial \Omega)$, is uniquely solvable in the class $L^p_{2-t}(\Omega)$ for $(t,\frac{1}{p})\in {\mathcal V}_{\epsilon}$, where ${\mathcal V}_{\epsilon}$ ($\epsilon\geq 0$) 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.