# 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 -

## References

top- A. Agrachev and M. Caponigro, Controllability on the group of diffeomorphisms. Preprint (2008). Zbl1188.93016
- J.H. Albert, Genericity of simple eigenvalues for elliptic PDE's. Proc. Amer. Math. Soc.48 (1975) 413–418. Zbl0302.35071
- W. Arendt and D. Daners, Uniform convergence for elliptic problems on varying domains. Math. Nachr.280 (2007) 28–49. Zbl1124.47055
- V.I. Arnol'd, Modes and quasimodes. Funkcional. Anal. i Priložen.6 (1972) 12–20.
- J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim.20 (1982) 575–597. Zbl0485.93015
- K. Beauchard, Y. Chitour, D. Kateb and R. Long, Spectral controllability for 2D and 3D linear Schrödinger equations. J. Funct. Anal.256 (2009) 3916–3976. Zbl1191.35222
- T. Chambrion, P. Mason, M. Sigalotti and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009) 329–349. Zbl1161.35049
- Y. Chitour, J.-M. Coron and M. Garavello, On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete Contin. Dyn. Syst.14 (2006) 643–672. Zbl1134.93313
- S. Cox and E. Zuazua, The rate at which energy decays in a damped string. Comm. Partial Differential Equations19 (1994) 213–243. Zbl0818.35072
- Y.C. de Verdière, Sur une hypothèse de transversalité d'Arnol'd. Comment. Math. Helv.63 (1988) 184–193. Zbl0672.58046
- P. Hébrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string. Systems Control Lett.48 (2003) 199–209. Zbl1134.93399
- P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim.44 (2005) 349–366 (electronic). Zbl1083.49002
- A. Henrot and M. Pierre, Variation et optimisation de formes, Mathématiques et Applications48. Springer-Verlag, Berlin (2005). Zbl1098.49001
- L. Hillairet and C. Judge, Generic spectral simplicity of polygons. Proc. Amer. Math. Soc.137 (2009) 2139–2145. Zbl1169.58007
- T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften132. Springer-Verlag New York, Inc., New York (1966).
- J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV1 (1995/1996) 1–15 (electronic). Zbl0878.93034
- J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial differential equations and applications, Lect. Notes Pure Appl. Math.177, Dekker, New York (1996) 221–235. Zbl0852.35112
- T.J. Mahar and B.E. Willner, Sturm-Liouville eigenvalue problems in which the squares of the eigenfunctions are linearly dependent. Comm. Pure Appl. Math.33 (1980) 567–578. Zbl0422.34024
- A.M. Micheletti, Metrica per famiglie di domini limitati e proprietà generiche degli autovalori. Ann. Scuola Norm. Sup. Pisa26 (1972) 683–694. Zbl0255.35028
- A.M. Micheletti, Perturbazione dello spettro dell'operatore di Laplace, in relazione ad una variazione del campo. Ann. Scuola Norm. Sup. Pisa. 26 (1972) 151–169.
- F. Murat and J. Simon, Étude de problèmes d'optimal design, Lecture Notes in Computer Sciences41. Springer-Verlag, Berlin (1976). Zbl0334.49013
- J.H. Ortega and E. Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim.39 (2000) 1585–1614 (electronic). Zbl0980.35118
- J.H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the Stokes system in two space dimensions. Adv. Differential Equations6 (2001) 987–1023. Zbl1223.35252
- J.H. Ortega and E. Zuazua, Addendum to: Generic simplicity of the spectrum and stabilization for a plate equation [SIAM J. Control Optim. 39 (2000) 1585–1614; mr1825594]. SIAM J. Control Optim.42 (2003) 1905–1910 (electronic). Zbl0980.35118
- J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics16. Springer-Verlag, Berlin (1992). Zbl0761.73003
- E.D. Sontag, Mathematical control theory: Deterministic finite-dimensional systems, Texts in Applied Mathematics6. Springer-Verlag, New York (1990). Zbl0703.93001
- M. Teytel, How rare are multiple eigenvalues? Comm. Pure Appl. Math.52 (1999) 917–934.
- K. Uhlenbeck, Generic properties of eigenfunctions. Amer. J. Math.98 (1976) 1059–1078. Zbl0355.58017
- E. Zuazua, Switching controls. J. Eur. Math. Soc. (to appear). Zbl1203.49011

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