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<p>We study the asymptotics of eigenvalues and eigenfunctions of the spectral problem for the Laplace operator in a planar domain with the third boundary condition on the boundary, where the (variable) Robin coefficient is negative and large.Known and new asymptotic formulas are presented for both negative and positive eigenvalues and for eigenfunctions, revealing various types of their localization.In addition to the formal asymptotic analysis and a brief review of earlier results, a procedure is described for justifying the asymptotics in the previously unexplored case of a constant Robin coefficient and a global degenerate maximum of the boundary curvature realized at several points.</p>

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Plurality of Asymptotic Series of Eigenvalues for the Third Boundary Value Problem with a Large Negative Robin Coefficient

  • 期刊论文

  • 发表日期：

    2025年11月24日

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  • S. A. Nazarov

摘要

We study the asymptotics of eigenvalues and eigenfunctions of the spectral problem for the Laplace operator in a planar domain with the third boundary condition on the boundary, where the (variable) Robin coefficient is negative and large.Known and new asymptotic formulas are presented for both negative and positive eigenvalues and for eigenfunctions, revealing various types of their localization.In addition to the formal asymptotic analysis and a brief review of earlier results, a procedure is described for justifying the asymptotics in the previously unexplored case of a constant Robin coefficient and a global degenerate maximum of the boundary curvature realized at several points.