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

Let n >= 3 and Omega be a bounded Lipschitz domain in R-n. Assume that p is an element of (2, infinity) and the function b is an element of L infinity(partial derivative Omega) is non-negative, where partial derivative Omega denotes the boundary of Omega. Denote by v the outward unit normal to partial derivative Omega. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation Delta u = 0 in Omega with boundary data partial derivative u/partial derivative v + bu = f is an element of L-P(partial derivative Omega), respectively, in terms of a weak reverse Holder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted L-2(partial derivative Omega) space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Omega with boundary data in (weighted) Lp (partial derivative Omega) for any given p is an element of (1, infinity). (C) 2017 Elsevier Inc. All rights reserved.