Level sets of potential functions bisecting unbounded quadrilaterals

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Nasser, Mohamed M.S.
Nasyrov, Semen
Vuorinen, Matti

Nasser, M.M.S., Nasyrov, S. & Vuorinen, M. Level sets of potential functions bisecting unbounded quadrilaterals. Anal.Math.Phys. 12, 149 (2022). https://doi.org/10.1007/s13324-022-00732-3


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(?) 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.

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