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.