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.
Mots-clés : 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://www.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://www.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://www.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, Tome 16 (2010) no. 3, pp. 794-805. doi : 10.1051/cocv/2009014. http://www.numdam.org/articles/10.1051/cocv/2009014/
[1] and , Controllability on the group of diffeomorphisms. Preprint (2008). | Zbl
[2] , Genericity of simple eigenvalues for elliptic PDE's. Proc. Amer. Math. Soc. 48 (1975) 413-418. | Zbl
[3] and , Uniform convergence for elliptic problems on varying domains. Math. Nachr. 280 (2007) 28-49. | Zbl
[4] , Modes and quasimodes. Funkcional. Anal. i Priložen. 6 (1972) 12-20. | Zbl
[5] , and , Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575-597. | Zbl
[6] , , and , Spectral controllability for 2D and 3D linear Schrödinger equations. J. Funct. Anal. 256 (2009) 3916-3976. | Zbl
[7] , , and , 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] , and , On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete Contin. Dyn. Syst. 14 (2006) 643-672. | Zbl
[9] and , The rate at which energy decays in a damped string. Comm. Partial Differential Equations 19 (1994) 213-243. | Zbl
[10] , Sur une hypothèse de transversalité d'Arnol'd. Comment. Math. Helv. 63 (1988) 184-193. | Zbl
[11] and , Optimal shape and position of the actuators for the stabilization of a string. Systems Control Lett. 48 (2003) 199-209. | Zbl
[12] and , A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim. 44 (2005) 349-366 (electronic). | Zbl
[13] and , Variation et optimisation de formes, Mathématiques et Applications 48. Springer-Verlag, Berlin (2005). | Zbl
[14] and , Generic spectral simplicity of polygons. Proc. Amer. Math. Soc. 137 (2009) 2139-2145. | Zbl
[15] , Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag New York, Inc., New York (1966). | Zbl
[16] and , Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995/1996) 1-15 (electronic). | Numdam | Zbl
[17] and , 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] and , Sturm-Liouville eigenvalue problems in which the squares of the eigenfunctions are linearly dependent. Comm. Pure Appl. Math. 33 (1980) 567-578. | Zbl
[19] , Metrica per famiglie di domini limitati e proprietà generiche degli autovalori. Ann. Scuola Norm. Sup. Pisa 26 (1972) 683-694. | Numdam | Zbl
[20] , 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] and , Étude de problèmes d'optimal design, Lecture Notes in Computer Sciences 41. Springer-Verlag, Berlin (1976). | Zbl
[22] and , Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim. 39 (2000) 1585-1614 (electronic). | Zbl
[23] and , Generic simplicity of the eigenvalues of the Stokes system in two space dimensions. Adv. Differential Equations 6 (2001) 987-1023.
[24] and , 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] and , Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992). | Zbl
[26] , Mathematical control theory: Deterministic finite-dimensional systems, Texts in Applied Mathematics 6. Springer-Verlag, New York (1990). | Zbl
[27] , How rare are multiple eigenvalues? Comm. Pure Appl. Math. 52 (1999) 917-934. | Zbl
[28] , Generic properties of eigenfunctions. Amer. J. Math. 98 (1976) 1059-1078. | Zbl
[29] , Switching controls. J. Eur. Math. Soc. (to appear). | Zbl
Cité par Sources :
