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

Yannick Privat (1, 2) , Mario Sigalotti (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 CORIDA - Robust control of infinite dimensional systems and applications

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.

Mots clés: Generic; Laplacian-Dirichlet eigenfunctions; non-resonant spectrum; shape optimization; control

Domaines: Optimisation et contrôle [math.OC]

Yannick Privat, Mario Sigalotti. The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent. 2009. ⟨hal-00322301v2⟩