The squares of the laplacian-Dirichlet eigenfunctions are generically linearly independent
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, p. 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.

DOI : https://doi.org/10.1051/cocv/2009014
Classification:  37C20,  47A55,  47A75,  49K20,  49K30,  93B05
Keywords: genericity, laplacian-Dirichlet eigenfunctions, non-resonant spectrum, shape optimization, control
@article{COCV_2010__16_3_794_0,
     author = {Privat, Yannick and Sigalotti, Mario},
     title = {The squares of the laplacian-Dirichlet eigenfunctions are generically linearly independent},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {3},
     year = {2010},
     pages = {794-805},
     doi = {10.1051/cocv/2009014},
     zbl = {1206.35181},
     mrnumber = {2674637},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2010__16_3_794_0}
}
Privat, Yannick; Sigalotti, Mario. The squares of the laplacian-Dirichlet eigenfunctions are generically linearly independent. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 794-805. doi : 10.1051/cocv/2009014. http://www.numdam.org/item/COCV_2010__16_3_794_0/

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