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Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities

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Abstract

In this paper we show how the method of parallel coordinates can be extended to three dimensions. As an application, we prove the conjecture of Antunes et al. (Adv Calc Var 10:357–380, 2017) that “the ball maximises the first Robin eigenvalue with negative boundary parameter among all convex domains of equal surface area” under the weaker restriction that the boundary of the domain is diffeomorphic to the sphere and convex or axiconvex. We also provide partial results in arbitrary dimensions.

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Acknowledgements

This work is supported by the Russian Science Foundation under Grant 18-11-00316 Geometric methods in non-linear problems of mathematical physics. The author is indebted to A. V. Penskoi who ispired the interest to spectral geometry for suggesting this problem. The author also is indebted to O. I. Mokhov for fruitful discussions and important remarks. The author is grateful to D. Krejčiřík and V. Lotoreichik for useful discussions and to the Czech Technical University for its hospitality.

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  1. L.D.Landau Institute for Theoretical Physics, pr. Ak Semenova 1a, Chernogolovka, 142432, Russia

    Anastasia V. Vikulova

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  1. Anastasia V. Vikulova
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Correspondence to Anastasia V. Vikulova.

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Vikulova, A.V. Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities. Ricerche mat 71, 41–52 (2022). https://doi.org/10.1007/s11587-020-00533-5

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  • DOI: https://doi.org/10.1007/s11587-020-00533-5

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