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

Yannick Privat; Mario Sigalotti

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 3, page 794-805
- ISSN: 1292-8119

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topPrivat, Yannick, and Sigalotti, Mario. "The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 794-805. <http://eudml.org/doc/252257>.

@article{Privat2010,

abstract = {
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.
},

author = {Privat, Yannick, Sigalotti, Mario},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Genericity; Laplacian-Dirichlet eigenfunctions; non-resonant spectrum; shape optimization; control; genericity},

language = {eng},

month = {7},

number = {3},

pages = {794-805},

publisher = {EDP Sciences},

title = {The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent},

url = {http://eudml.org/doc/252257},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Privat, Yannick

AU - Sigalotti, Mario

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/7//

PB - EDP Sciences

VL - 16

IS - 3

SP - 794

EP - 805

AB -
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.

LA - eng

KW - Genericity; Laplacian-Dirichlet eigenfunctions; non-resonant spectrum; shape optimization; control; genericity

UR - http://eudml.org/doc/252257

ER -

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