Publication ahead of print Issue ESAIM: COCV DOI 10.1051/cocv/2009014 Published online 02 July 2009

ESAIM: COCV
DOI: 10.1051/cocv/2009014

The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

Yannick Privat¹ and Mario Sigalotti<sup>1, 2

1</sup>  Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France.
²  INRIA Nancy – Grand Est, France. Mario.sigalotti@inria.fr

Received September 17, 2008. Revised March 11, 2009. Published online July 2nd, 2009.

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.

Mathematics Subject Classification. 37C20, 47A55, 47A75, 49K20, 49K30, 93B05

Key words: Genericity, Laplacian-Dirichlet eigenfunctions, non-resonant spectrum, shape optimization, control


© EDP Sciences, SMAI 2009

What is OpenURL?