Solution of the Robin and Dirichlet Problem for the Laplace Equation

Abstract

Suppose that G ⊂ R ^m (m ≥ 2) is an open set with a non-void compact boundary ∂G such that ∂G = ∂(cl G), where cl G is the closure of G. Fix a nonnegative element λ of C′(∂G) (=the Banach space of all finite signed Borel measures with support in ∂G with the total variation as a norm) and suppose that the single layer potential uλ is bounded and continuous on ∂G. (In R ² it means that λ = 0. If G ⊂ R ^m, (m > 2), ∂G is locally Lipschitz, λ = fℋ, ℋ is the surface measure on the boundary of G, f is a nonnegative bounded measurable function, then uλ is bounded and continuous.)Here

$$\mu \nu (x)\, = \,\int\limits_{{R^m}} {{h_x}(y)\,d\nu (y)} ,$$

where ν ∈ C’(∂G),

$${h_x}(y)\, = \,\left\{ {\begin{array}{*{20}{c}} {{{(m - 2)}^{ - 1}}{A^{ - 1}}{{\left| {x - y} \right|}^{2 - m}},\,m > 2,} \\ {{A^{ - 1}}\log {{\left| {x - y} \right|}^{ - 1}},m = 2,} \end{array}} \right.$$

A is the area of the unit sphere in R^m.