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laplacian-dirichlet eigenfunctions

ESAIM: Control, Optimisation and Calculus of Variations (1)

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The squares of the Laplacian-Dirichlet eigenfunctionsare generically linearly independent
Yannick Privat, Mario Sigalotti
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 16 / Issue 3 / July 2010

Published online by Cambridge University Press:

02 July 2009, pp. 794-805

Print publication:

July 2010
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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.