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Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues

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Abstract

We study the convergence of extrema of averages of eigenvalues of the Dirichlet Laplacian on domains in \(\mathbb {R}^{n}\) under both measure and surface measure restrictions. In the former case we prove that the sequence of averages to the power n / 2 is sub-additive and determine the first term in its asymptotics in the high-frequency limit. In the latter case, we show that the sequence of minimisers converges to the ball as the frequency goes to infinity. Similar results hold for Neumann boundary conditions.

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Acknowledgements

This research was partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Pedro Freitas

  2. Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016, Lisboa, Portugal

    Pedro Freitas

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  1. Pedro Freitas
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Correspondence to Pedro Freitas.

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In memoriam Ahmad El Soufi.

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Freitas, P. Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues. J Stat Phys 167, 1511–1518 (2017). https://doi.org/10.1007/s10955-017-1789-8

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  • DOI: https://doi.org/10.1007/s10955-017-1789-8

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