The method of integral equations in the generalized jump problem for the Laplace equation outside cuts on the plane
There is considered a boundary value problem for harmonic functions outside a finite number of cuts in the plane. Let \(\Gamma\) denote the union of the cuts, then the boundary condition has the form \[ \begin{aligned} ( u^+(x) - g(s) u^-(x)) |_{x(s) \in \Gamma}&= f_1(s), \\ \Biggl( g(s) \biggl( \frac{ \partial u}{\partial n_x} \biggr)^+ - \biggl( \frac{ \partial u}{\partial n_x} \biggr)^- \Biggr) |_{x(s) \in \Gamma}= f_2(s), \end{aligned} \] where \(g, f_1, f_2\) are given functions, and the behavior at infinity is governed by the condition: for \(|x |\to \infty\) there holds \[ u(x) = A \ln |x|+ {\mathcal O}(1),\qquad \frac{ \partial u}{\partial |x|} = \frac{A}{|x|} + {\mathcal O}\left( \frac{1}{|x|^2} \right), \] with \(A=\)const. It is proved that the problem has at most one solution; it is shown that the boundary value problem reduces to a system of singular integral equations (with the Cauchy kernel) the latter having always a solution.
scientific article; zbMATH DE number 1916173