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Isoperimetric control of the Steklov spectrum

Abstract : We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.
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https://hal.archives-ouvertes.fr/hal-00779282
Contributor : Ahmad El Soufi <>
Submitted on : Tuesday, January 22, 2013 - 12:00:02 AM
Last modification on : Thursday, November 29, 2018 - 1:19:46 AM
Document(s) archivé(s) le : Tuesday, April 23, 2013 - 3:51:23 AM

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  • HAL Id : hal-00779282, version 1

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Bruno Colbois, Ahmad El Soufi, Alexandre Girouard. Isoperimetric control of the Steklov spectrum. Journal of Functional Analysis, Elsevier, 2011, 261 (5), pp.1384-1399. ⟨hal-00779282⟩

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