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 potential function of the quadrilateral. We compute the values of the potential function u including its value at infinity. The main result of this paper is Theorem 4.3 which yields a formula for u(∞) expressed in terms of the angles of the polygonal given quadrilateral and the well-known special functions. We use two independent numerical methods to illustrate our result. The first method is a Mathematica program and the second one is based on using the MATLAB toolbox PlgCirMap. The case of a quadrilateral, which is the exterior of the unit disc with four fixed points on its boundary, is considered as well.