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The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Auteur(s)
Hassannezhad, Asma
Date de parution
2020-4-1
In
Journal of Functional Analysis
Vol.
6
No
278
De la page
1
A la page
32
Revu par les pairs
1
Résumé
Given two compact Riemannian manifolds $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighbourhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant $C$ such that $
\sigma_k(M_1)-\sigma_k(M_2)
\leq C$ for each $k\in\N$. The constant $C$ depends only on the geometry of $\Omega_1\cong\Omega_2$. This follows from a quantitative relationship between the Steklov eigenvalues $\sigma_k$ of a compact Riemannian manifold $M$ and the eigenvalues $\lambda_k$ of the Laplacian on
its boundary. Our main result states that the difference $
its boundary. Our main result states that the difference $
\sigma_k-\sqrt{\lambda_k}
$ is bounded above by a constant which depends on the geometry of $M$ only in a neighbourhood of its boundary.
The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant $C$ is given explicitly in terms of bounds on the geometry of $\Omega_1\cong\Omega_2$.
The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant $C$ is given explicitly in terms of bounds on the geometry of $\Omega_1\cong\Omega_2$.
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Type de publication
journal article
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