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Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter

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Abstract

This paper is motivated by the study of the existence of optimal domains maximizing the kth Robin Laplacian eigenvalue among sets of prescribed measure, in the case of a negative boundary parameter. We answer positively to this question and prove an existence result in the class of measurable sets and for quite general spectral functionals. The key tools of our analysis rely on tight isodiametric and isoperimetric geometric controls of the eigenvalues. In two dimensions of the space, under simply connectedness assumptions, further qualitative properties are obtained on the optimal sets.

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Notes

  1. This remark is due to James Kennedy.

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Acknowledgements

The first author was supported by the “Geometry and Spectral Optimization” research programme LabEx PERSYVAL-Lab GeoSpec (ANR-11-LABX-0025-01) and ANR SHAPO “Optimisation de forme” (ANR-18-CE40-0013).

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Authors and Affiliations

  1. Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000, Chambéry, France

    Dorin Bucur

  2. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100, Lecce, Italy

    Simone Cito

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  1. Dorin Bucur
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  2. Simone Cito
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Correspondence to Dorin Bucur.

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Dorin Bucur is member of the Institut Universitaire de France.

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Bucur, D., Cito, S. Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter. J Geom Anal 30, 4356–4385 (2020). https://doi.org/10.1007/s12220-019-00245-9

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  • DOI: https://doi.org/10.1007/s12220-019-00245-9

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