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Level sets of potential functions bisecting unbounded quadrilaterals

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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 \(u(\infty )\) 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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Funding

The work of the second author is performed under the development program of Volga Region Mathematical Center (agreement no. 075-02-2022-882). The first and third authors did not receive support from any organization for the submitted work.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS, 67260-0033, USA

    Mohamed M. S. Nasser

  2. Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar

    Mohamed M. S. Nasser

  3. Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, 420008, Russia

    Semen Nasyrov

  4. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Matti Vuorinen

Authors
  1. Mohamed M. S. Nasser
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  2. Semen Nasyrov
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  3. Matti Vuorinen
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Correspondence to Semen Nasyrov.

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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

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  • DOI: https://doi.org/10.1007/s13324-022-00732-3

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