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

Given a bounded Lipschitz domain Ω ⊂ ℝⁿ n ≥ 3, we prove that the Poisson's problem for the Laplacian with right-hand side in L_-t^p(Ω), Robin-type boundary datum in the Besov space B_p^1-1/p-t,p(∂Ω) and non-negative, non-everywhere vanishing Robin coefficient b ∈ L ^n-1(∂ω), is uniquely solvable in the class L _2-t^p(Omega;) for (t, 1/p) ∈ ν_ε, where ν_ε (ε ≥ 0) is an open (Ω,b)-dependent plane region and ν₀ 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.