Level sets of potential functions bisecting unbounded quadrilaterals

TL;DR

This paper investigates the potential functions associated with unbounded quadrilaterals, solving a mixed boundary value problem for the Laplace equation to understand harmonic functions in these geometric configurations.

Contribution

It provides explicit computations of the potential function values, including at infinity, for quadrilaterals with mixed boundary conditions, advancing understanding of harmonic functions in polygonal domains.

Findings

Explicit potential function values computed for unbounded quadrilaterals

Analysis of harmonic functions with mixed boundary conditions

Insights into boundary value problems in polygonal domains

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.