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.

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: 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},
     pages = {794--805},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {3},
     year = {2010},
     doi = {10.1051/cocv/2009014},
     mrnumber = {2674637},
     zbl = {1206.35181},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2009014/}
}
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
PY  - 2010
SP  - 794
EP  - 805
VL  - 16
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2009014/
DO  - 10.1051/cocv/2009014
LA  - en
ID  - COCV_2010__16_3_794_0
ER  - 
%0 Journal Article
%A Privat, Yannick
%A Sigalotti, Mario
%T The squares of the laplacian-Dirichlet eigenfunctions are generically linearly independent
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2010
%P 794-805
%V 16
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2009014/
%R 10.1051/cocv/2009014
%G en
%F 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://archive.numdam.org/articles/10.1051/cocv/2009014/

[1] A. Agrachev and M. Caponigro, Controllability on the group of diffeomorphisms. Preprint (2008). | Zbl

[2] J.H. Albert, Genericity of simple eigenvalues for elliptic PDE's. Proc. Amer. Math. Soc. 48 (1975) 413-418. | Zbl

[3] W. Arendt and D. Daners, Uniform convergence for elliptic problems on varying domains. Math. Nachr. 280 (2007) 28-49. | Zbl

[4] V.I. Arnol'D, Modes and quasimodes. Funkcional. Anal. i Priložen. 6 (1972) 12-20. | Zbl

[5] J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575-597. | Zbl

[6] 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. | Zbl

[7] 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 Linéaire 26 (2009) 329-349. | Numdam | Zbl

[8] 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. | Zbl

[9] S. Cox and E. Zuazua, The rate at which energy decays in a damped string. Comm. Partial Differential Equations 19 (1994) 213-243. | Zbl

[10] Y.C. De Verdière, Sur une hypothèse de transversalité d'Arnol'd. Comment. Math. Helv. 63 (1988) 184-193. | Zbl

[11] 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. | Zbl

[12] P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim. 44 (2005) 349-366 (electronic). | Zbl

[13] A. Henrot and M. Pierre, Variation et optimisation de formes, Mathématiques et Applications 48. Springer-Verlag, Berlin (2005). | Zbl

[14] L. Hillairet and C. Judge, Generic spectral simplicity of polygons. Proc. Amer. Math. Soc. 137 (2009) 2139-2145. | Zbl

[15] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag New York, Inc., New York (1966). | Zbl

[16] J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995/1996) 1-15 (electronic). | Numdam | Zbl

[17] 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. | Zbl

[18] 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. | Zbl

[19] A.M. Micheletti, Metrica per famiglie di domini limitati e proprietà generiche degli autovalori. Ann. Scuola Norm. Sup. Pisa 26 (1972) 683-694. | Numdam | Zbl

[20] 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. | Numdam | Zbl

[21] F. Murat and J. Simon, Étude de problèmes d'optimal design, Lecture Notes in Computer Sciences 41. Springer-Verlag, Berlin (1976). | Zbl

[22] 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). | Zbl

[23] J.H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the Stokes system in two space dimensions. Adv. Differential Equations 6 (2001) 987-1023.

[24] 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). | Zbl

[25] J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992). | Zbl

[26] E.D. Sontag, Mathematical control theory: Deterministic finite-dimensional systems, Texts in Applied Mathematics 6. Springer-Verlag, New York (1990). | Zbl

[27] M. Teytel, How rare are multiple eigenvalues? Comm. Pure Appl. Math. 52 (1999) 917-934. | Zbl

[28] K. Uhlenbeck, Generic properties of eigenfunctions. Amer. J. Math. 98 (1976) 1059-1078. | Zbl

[29] E. Zuazua, Switching controls. J. Eur. Math. Soc. (to appear). | Zbl

Cited by Sources: