Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

Wolfgang Arendt; Rafe Mazzeo

doi:10.3934/cpaa.2012.11.2201

Article Contents

2012, Volume 11, Issue 6: 2201-2212. Doi: 10.3934/cpaa.2012.11.2201

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Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

Wolfgang Arendt^1, and
Rafe Mazzeo^2,

1.
Abteilung Angewante Analysis, Universität Ulm, 89069 Ulm
2.
Department of Mathematics, Stanford University, Stanford, CA 94305

Received: January 31, 2011

Revised: January 31, 2011

Published: October 2012

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Abstract

If $\Omega$ is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary $\Gamma = \partial \Omega$, the Dirichlet-to-Neumann operator $\mathcal{D}_\lambda$ is defined on $L^2(\Gamma)$ for any real $\lambda$. We prove a close relationship between the eigenvalues of $\mathcal{D}_\lambda$ and those of the Robin Laplacian $\Delta_\mu$, i.e. the Laplacian with Robin boundary conditions $\partial_\nu u =\mu u$. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, $\lambda^N_{k+1} \leq \lambda^D_k$, $k \in N$, and to sharpen the inequality to be strict, whenever $\Omega$ is a Lipschitz domain in $R^d$. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by $-\mathcal{D}_\lambda$, for $\lambda$ sufficiently small or negative, is irreducible.

Keywords:

Eigenvalue inequalities,
Dirichlet-to-Neumann operator.

Mathematics Subject Classification: Primary: 35P15; Secondary: 47D06.

Citation:

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