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Double- and Simple-Layer Potentials for a Three-Dimensional Elliptic Equation with a Singular Coefficient and Their Applications

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Abstract

The double- and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. The desired solution represents a potential of a certain layer with unknown density. One finds it with the help of the theory of Fredholm integral equations of the second kind. The potential, in turn, is expressed via the fundamental solution to the given elliptic equation. At present, fundamental solutions to Helmholtz multidimensional singular equations are known, nevertheless, the potential theory is constructed only for two-dimensional degenerate equations. In this paper, we study the mentioned potentials for a three-dimensional elliptic equation with one singular coefficient and apply the obtained results to solving the Dirichlet problem.

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Authors and Affiliations

  1. V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 81 Mirzo Ulugbek str., 100170, Tashkent, Republic of Uzbekistan

    T. G. Ergashev

  2. Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39 Kari Niyazi str., 100000, Tashkent, Republic of Uzbekistan

    T. G. Ergashev

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  1. T. G. Ergashev
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Correspondence to T. G. Ergashev.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 1, pp. 81–96.

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Ergashev, T.G. Double- and Simple-Layer Potentials for a Three-Dimensional Elliptic Equation with a Singular Coefficient and Their Applications. Russ Math. 65, 72–86 (2021). https://doi.org/10.3103/S1066369X21010060

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  • DOI: https://doi.org/10.3103/S1066369X21010060

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