Solution of the Robin and Dirichlet problem for the Laplace equation

Suppose that G subset of R-m (m greater than or equal to 2) is an open set with a non-void compact boundary partial derivativeG such that partial derivativeG = partial derivative (cl G), where cl G is the closure of G. Fix a nonnegative element lambda of C'(partial derivativeG) (=the Banach space of all finite signed Borel measures with support in partial derivativeG with the total variation as a norm) and suppose that the single layer potential U lambda is bounded and continuous on partial derivativeG. (In R-2 it means that lambda = 0. If G subset of R-m, (m > 2), partial derivativeG is locally Lipschitz, lambda = fH, H is the surface measure on the boundary of G, f is a nonnegative bounded measurable function, then U lambda is bounded and continuous.)Here U nu (x) = integral (Rm) h(x)(y)d nu (y), where nu is an element of C'(partial derivativeG), h(x)(y) = {((m-2)-1A-1\x-y\2-mm n>2m,)(A-1log\x-y\-1, m=2) A is the area of the unit sphere in R-m.