Solution of the Robin problem for the Laplace equation
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 \(\lambda (\partial H)=0\).
scientific article; zbMATH DE number 1450147
scientific article