Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T05:38:40.974Z Has data issue: false hasContentIssue false
  • English
  • Français

The squares of the Laplacian-Dirichlet eigenfunctionsare generically linearly independent

Published online by Cambridge University Press:  02 July 2009

Yannick Privat  and
Mario Sigalotti
Yannick Privat
Affiliation:
Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœ uvre-lès-Nancy Cedex, France.
Mario Sigalotti
Affiliation:
Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœ uvre-lès-Nancy Cedex, France. INRIA Nancy – Grand Est, France. [email protected]
Get access

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.

Keywords

GenericityLaplacian-Dirichlet eigenfunctionsnon-resonant spectrumshape optimizationcontrol
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 794 - 805
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Agrachev and M. Caponigro, Controllability on the group of diffeomorphisms. Preprint (2008).
Albert, J.H., Genericity of simple eigenvalues for elliptic PDE's. Proc. Amer. Math. Soc. 48 (1975) 413–418.
Arendt, W. and Daners, D., Uniform convergence for elliptic problems on varying domains. Math. Nachr. 280 (2007) 28–49. CrossRef
Arnol'd, V.I., Modes and quasimodes. Funkcional. Anal. i Priložen. 6 (1972) 12–20.
Ball, J.M., Marsden, J.E. and Slemrod, M., Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575–597. CrossRef
Beauchard, K., Chitour, Y., Kateb, D. and Long, R., Spectral controllability for 2D and 3D linear Schrödinger equations. J. Funct. Anal. 256 (2009) 3916–3976. CrossRef
Chambrion, T., Mason, P., Sigalotti, M. and Boscain, U., Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 329–349. CrossRef
Chitour, Y., Coron, J.-M. and Garavello, M., On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete Contin. Dyn. Syst. 14 (2006) 643–672.
Cox, S. and Zuazua, E., The rate at which energy decays in a damped string. Comm. Partial Differential Equations 19 (1994) 213–243. CrossRef
de Verdière, Y.C., Sur une hypothèse de transversalité d'Arnol'd. Comment. Math. Helv. 63 (1988) 184–193. CrossRef
Hébrard, P. and Henrot, A., Optimal shape and position of the actuators for the stabilization of a string. Systems Control Lett. 48 (2003) 199–209. CrossRef
Hébrard, P. and Henrot, A., A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim. 44 (2005) 349–366 (electronic). CrossRef
A. Henrot and M. Pierre, Variation et optimisation de formes, Mathématiques et Applications 48. Springer-Verlag, Berlin (2005).
Hillairet, L. and Judge, C., Generic spectral simplicity of polygons. Proc. Amer. Math. Soc. 137 (2009) 2139–2145. CrossRef
T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag New York, Inc., New York (1966).
J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995/1996) 1–15 (electronic).
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.
Mahar, T.J. and Willner, B.E., Sturm-Liouville eigenvalue problems in which the squares of the eigenfunctions are linearly dependent. Comm. Pure Appl. Math. 33 (1980) 567–578. CrossRef
Micheletti, A.M., Metrica per famiglie di domini limitati e proprietà generiche degli autovalori. Ann. Scuola Norm. Sup. Pisa 26 (1972) 683–694.
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 Sciences 41. Springer-Verlag, Berlin (1976).
Ortega, J.H. and Zuazua, E., Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim. 39 (2000) 1585–1614 (electronic). CrossRef
Ortega, J.H. and Zuazua, E., Generic simplicity of the eigenvalues of the Stokes system in two space dimensions. Adv. Differential Equations 6 (2001) 987–1023.
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).
J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992).
E.D. Sontag, Mathematical control theory: Deterministic finite-dimensional systems, Texts in Applied Mathematics 6. Springer-Verlag, New York (1990).
Teytel, M., How rare are multiple eigenvalues? Comm. Pure Appl. Math. 52 (1999) 917–934. 3.0.CO;2-S>CrossRef
Uhlenbeck, K., Generic properties of eigenfunctions. Amer. J. Math. 98 (1976) 1059–1078. CrossRef
E. Zuazua, Switching controls. J. Eur. Math. Soc. (to appear).