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The squares of the Laplacian-Dirichlet eigenfunctionsare generically linearly independent

Published online by Cambridge University Press:  02 July 2009

Yannick Privat  and
Mario Sigalotti
Yannick Privat
Affiliation:
Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœ uvre-lès-Nancy Cedex, France.
Mario Sigalotti
Affiliation:
Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœ uvre-lès-Nancy Cedex, France. INRIA Nancy – Grand Est, France. Mario.sigalotti@inria.fr
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Abstract

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties.The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schrödinger equation.

Keywords

GenericityLaplacian-Dirichlet eigenfunctionsnon-resonant spectrumshape optimizationcontrol
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 794 - 805
Copyright
© EDP Sciences, SMAI, 2009

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