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Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions

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Abstract

This work, which can be considered as a small monograph, is devoted to the study of two-and three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of compatibility of asymptotic expansions, the boundary layer method, and the multiscale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very essential weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

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Authors and Affiliations

  1. Institute of Mathematics with Computer Center of the Ufa Science Center of the Russian Academy of Sciences, Ufa, Russia

    D. I. Borisov

  2. Bashkir State University, Ufa, Russia

    D. I. Borisov

  3. Bashkir State Pedagogical University named after M. Akmullah, Ufa, Russia

    D. I. Borisov

  4. University of Hradec Králové, Hradec Králové, Czech Republic

    D. I. Borisov

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 1, Partial Differential Equations, 2021.

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Borisov, D.I. Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions. J Math Sci 277, 841–958 (2023). https://doi.org/10.1007/s10958-023-06893-1

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