# Solution of the Robin and Dirichlet Problem for the Laplace Equation

• Dagmar Medková Praha
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)

## Abstract

Suppose that GR 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 GR 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 Rm.

## Keywords

Harmonic Function Dirichlet Problem Laplace Equation Potential Theory Neumann Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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