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A Faber–Krahn Inequality for Indented and Cut Membranes

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In 1960, Payne and Weinberger proved that, among all domains that lie within a wedge (an angle whose measure is less than or equal to π) and have a given value of a certain integral, the circular sector has the lowest fundamental eigenvalue of the Dirichlet Laplacian. We show that an analogue of this assertion is true for domains with a cut and for indented domains, i.e., for those located in a reflex angle (its measure is between π and 2π).

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Authors and Affiliations

  1. Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, RAS, V.O., Bol’shoy pr. 61, St Petersburg, 199178, Russia

    N. Kuznetsov

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  1. N. Kuznetsov
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Correspondence to N. Kuznetsov.

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Translated from Problemy Matematicheskogo Analiza 125, 2023, pp. 105-109.

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Kuznetsov, N. A Faber–Krahn Inequality for Indented and Cut Membranes. J Math Sci 276, 111–116 (2023). https://doi.org/10.1007/s10958-023-06728-z

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  • DOI: https://doi.org/10.1007/s10958-023-06728-z

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