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Dirichlet problem for the Helmholtz equation in an exterior two-dimensional domain with cuts without matching conditions at the cut endpoints

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Abstract

The Dirichlet problem for the Helmholtz equation in a plane exterior domain with cuts is considered for the case in which functions defined on opposite sides of the cuts in the Dirichlet boundary condition do not necessarily satisfy the matching conditions at the cut endpoints and the solution of the problem is not necessarily continuous at the endpoints of the cuts. We give a well-posed statement of the problem, prove existence and uniqueness theorems for a classical solution, derive an integral representation of the solution, and use it to study its properties. We show that the Dirichlet problem in the considered setting does not necessarily have a weak solution, although there exists a classical solution. We derive asymptotic formulas describing the behavior of the gradient of the solution at the endpoints of the cuts.

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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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Correspondence to P. A. Krutitskii.

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Original Russian Text © P.A. Krutitskii, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 9, pp. 1150–1163.

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Krutitskii, P.A. Dirichlet problem for the Helmholtz equation in an exterior two-dimensional domain with cuts without matching conditions at the cut endpoints. Diff Equat 50, 1136–1149 (2014). https://doi.org/10.1134/S001226611409002X

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  • DOI: https://doi.org/10.1134/S001226611409002X

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