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Solution of the Robin problem for the Laplace equation. (English) Zbl 0938.31005
Let $$G\subset \mathbb {R}^m$$ ($$m>2$$) be an open set with compact boundary $$\partial G\neq \emptyset$$. Let $$\lambda$$ be a finite Borel measure supported by $$\partial G$$ and suppose that the single layer potential $$U\lambda$$ of the measure $$\lambda$$ is continuous and bounded on $$\partial G$$. Further let $$\mu$$ be a finite signed Borel measure supported by $$\partial G$$. A generalized Robin problem for the Laplace equation on $$G$$ is formulated in the following way: Find a function $$u\in L^1(\lambda)$$ on $$\operatorname {cl}G$$ such that
(a) $$u$$ is harmonic on $$G$$,
(b) $$u$$ is finely continuous in $$\lambda$$-a.a. points of $$\partial G$$
(c) $$|\nabla u|$$ is integrable over all bounded open subsets of $$G$$,
(d) $$N^Gu+u\lambda =\mu$$, where $$N^Gu$$ denotes the weak characterization of the normal derivative of $$u$$.
The solution of that problem is represented by the single layer potential $$U\nu$$ of a signed measure $$\nu$$ on $$\partial G$$. It is proved that if $$G$$ has a smooth boundary or $$m=3$$ and $$G$$ has a piecewise-smooth boundary then there is a solution of that problem with given $$\mu$$ if and only if $$\mu (\partial H)=0$$ for all bounded components $$H$$ of $$\operatorname {cl}G$$ for which $$\lambda (\partial H)=0$$.
Reviewer: M.Dont (Praha)

##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J25 Boundary value problems for second-order elliptic equations
##### Keywords:
Laplace equation; Robin problem; single layer potential
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