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

Privat, Yannick; Sigalotti, Mario

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 794-805.

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.

Classification : 37C20, 47A55, 47A75, 49K20, 49K30, 93B05
Mots clés : genericity, laplacian-Dirichlet eigenfunctions, non-resonant spectrum, shape optimization, control

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