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

Abstract

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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.

How to cite

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Privat, 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 -

References

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