| Issue |
ESAIM: COCV
Volume 16, Number 3, July-September 2010
|
|
|---|---|---|
| Page(s) | 794 - 805 | |
| DOI | https://doi.org/10.1051/cocv/2009014 | |
| Published online | 02 July 2009 | |
The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
1
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.sigalotti@inria.fr
Received:
17
September
2008
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
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