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Generalization of the Neumann problem to harmonic functions outside cuts on the plane

  • Partial Differential Equations
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Abstract

We consider a boundary value problem for harmonic functions outside cuts on the plane. The jump of the normal derivative and a linear combination of the normal derivative on one side with the jump of the unknown function are given on each cut. The problem is considered with three conditions at infinity, which lead to distinct results on the existence and number of solutions. We obtain an integral representation of the solution in the form of potentials whose density satisfies a uniquely solvable Fredholm integral equation of the second kind.

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Authors and Affiliations

  1. Institute for Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    P. A. Krutitskii

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  1. P. A. Krutitskii
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Original Russian Text © P.A. Krutitskii, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 9, pp. 1100–1112.

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Krutitskii, P.A. Generalization of the Neumann problem to harmonic functions outside cuts on the plane. Diff Equat 49, 1062–1074 (2013). https://doi.org/10.1134/S0012266113090024

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