The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

This article has an erratum: [erratum]

Issue		ESAIM: COCV Volume 16, Number 3, July-September 2010


Page(s)		794 - 805
DOI		10.1051/cocv/2009014
Published online		02 July 2009

ESAIM: COCV 16 (2010) 794-805

The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

Yannick Privat¹ and Mario Sigalotti¹^,2

¹ Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université – INRIA – CNRS, B.P. 239, 54506 Vandœ uvre-lès-Nancy Cedex, France.
² INRIA Nancy – Grand Est, France. Mario.sigalotti@inria.fr

Received: 17 September 2008
Revised: 11 March 2009

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.

Mathematics Subject Classification: 37C20 / 47A55 / 47A75 / 49K20 / 49K30 / 93B05.

Key words: Genericity / Laplacian-Dirichlet eigenfunctions / non-resonant spectrum / shape optimization / control.

What is OpenURL?

The OpenURL standard is a protocol for transmission of metadata describing the resource that you wish to access. An OpenURL link contains article metadata and directs it to the OpenURL server of your choice. The OpenURL server can provide access to the resource and also offer complementary services (specific search engine, export of references...). The OpenURL link can be generated by different means.

If your librarian has set up your subscription with an OpenURL resolver, OpenURL links appear automatically on the abstract pages.
You can define your own OpenURL resolver with your EDPS Account. In this case your choice will be given priority over that of your library.
You can use an add-on for your browser (Firefox or I.E.) to display OpenURL links on a page (see http://www.openly.com/openurlref/). You should disable this module if you wish to use the OpenURL server that you or your library have defined.