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The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change

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Abstract

In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.

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  1. Ufa State Petroleum Technological University, 450064, Ufa, Russia

    A. Gimaltdinova

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  1. A. Gimaltdinova
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Correspondence to A. Gimaltdinova.

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(Submitted by F. G. Avkhadiev)

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Gimaltdinova, A. The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change. Lobachevskii J Math 41, 2155–2167 (2020). https://doi.org/10.1134/S1995080220110098

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