Level sets of potential functions bisecting unbounded quadrilaterals

Abstract

We study the mixed Dirichlet-Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet\,/\,Neumann conditions at opposite pairs of sides are $\{0,1\}$ and $\{0,0\},$ resp. The solution to this problem is a harmonic function in the unbounded complement of the polygon known as the \emph{potential function} of the quadrilateral. We compute the values of the potential function including its value at infinity.