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Hypersurfaces with Prescribed Boundary and Small Steklov Eigenvalues

Part of: Calculus on manifolds; nonlinear operators Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  12 December 2019

Bruno Colbois ,
Alexandre Girouard  and
Antoine Métras
Bruno Colbois
Affiliation:
Université de Neuchâtel, Institut de Mathématiques, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland Email: [email protected]
Alexandre Girouard
Affiliation:
Département de mathématiques et de statistique, Université Laval, Pavillon AlexandreVachon, 1045, av. de la Médecine, Québec Qc G1V 0A6, Canada Email: [email protected]
Antoine Métras
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada Email: [email protected]
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Abstract

Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x2202}M$, we prove the existence of a sequence $M_{j}$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\unicode[STIX]{x1D70E}_{k}(M_{j})$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_{j}$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.

Keywords

Steklov eigenvaluehypersurface

MSC classification

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds
Secondary: 58C40: Spectral theory; eigenvalue problems
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 46 - 57
Copyright
© Canadian Mathematical Society 2019

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