Refereed journal article or data article (A1)

Level sets of potential functions bisecting unbounded quadrilaterals

List of Authors: Nasser Mohamed M.S, Nasyrov Semen, Vuorinen Matti

Publisher: SPRINGER BASEL AG

Publication year: 2022

Journal: Analysis and Mathematical Physics

Journal name in source: ANALYSIS AND MATHEMATICAL PHYSICS

Journal acronym: ANAL MATH PHYS

Volume number: 12

Number of pages: 15

ISSN: 1664-2368

DOI: http://dx.doi.org/10.1007/s13324-022-00732-3

URL: http://dx.doi.org/10.1007%2Fs13324-022-00732-3

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*(oo) 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.