Regularity of solutions of the Neumann problem for the Laplace equation

Abstract

Let u be a solution of the Neumann problem for the Laplace equation in G with the boundary condition g. It is shown that u ∈ L ^q (∂ G ) (equivalently, u ∈ B^q,2_1/q (G ) for 1 < q ≤ 2, u ∈ L^q _1/q (G ) for 2 ≤ q < ∞) if and only if the single layer potential corresponding to the boundary condition g  is in L ^q (∂ G ). As a consequence we give a regularity result for some nonlinear boundary value problem.