REGULARITY OF SOLUTIONS OF THE NEUMANN PROBLEM FOR THE LAPLACE EQUATION

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 is an element of L-q(partial derivative G) (equivalently, u is an element of B-1/q(q,2) (G) for 1 < q <= 2, u is an element of L-1/q(q)(G) for 2 <= q < infinity) if and only if the single layer potential corresponding to the boundary condition g is in L-q(partial derivative G. As a consequence we give a regularity result for some nonlinear boundary value problem.