Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues

Bruno Colbois; Alexandre Girouard; Antoine Métras

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arXiv · 1811.11463

Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues

Bruno Colbois·Alexandre Girouard·Antoine Métras

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Abstract

Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ 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.

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BibTeX RIS

Bruno Colbois, Alexandre Girouard, Antoine Métras. 2018-11-28. Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues. https://arxiv.org/abs/1811.11463

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