The Lp Robin problem for Laplace equations in Lipschitz and (semi-)convex domains

Let n≥3 and Ω be a bounded Lipschitz domain in Rn. Assume that p∈(2,∞) and the function b∈L∞(∂Ω) is non-negative, where ∂Ω denotes the boundary of Ω. Denote by ν the outward unit normal to ∂Ω. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation Δu=0 in Ω with boundary data ∂u/∂ν+bu=f∈Lp(∂Ω), respectively, in terms of a weak reverse Hölder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted L2(∂Ω) space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Ω with boundary data in (weighted) Lp(∂Ω) for any given p∈(1,∞).