The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
Institut Élie Cartan de Nancy, UMR 7502
Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœ
uvre-lès-Nancy Cedex, France.
2 INRIA Nancy – Grand Est, France. Mario.email@example.com
Revised: 11 March 2009
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