Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions
Consider Laplace's equation on a polygonal domain with homogeneous oblique boundary conditions cos \(\theta\) \({}_ k u_ n(z)+\sin \theta_ k u_{\sigma}(z)=0\) on the edges \(\Gamma_ k\) of the polygon. If at the corners neither regularity nor boundedness is required, the solution space has infinite dimension. First, it is shown that every solution is the real part of an analytic function which maps each \(\Gamma_ k\) onto a straight line segment oriented at the angle \(\theta_ k\). Then the exact forms of the general solution are given, first, under the assumption of at most polynomial growth near the vertices and, second, under the assumption of boundedness of continuity. In both cases f is a composition of an ordinary Schwarz-Christoffel map for the polygon and a similar function obtained by solving a generalized Schwarz-Christoffel parameter problem. Two applications are treated in detail: The Hall effect on a polygonal conductor and a Brownian motion problem with oblique reflection, which comes up in queueing theory.
scientific article; zbMATH DE number 3959863
scientific article