On the determination of Dirichlet or transmission eigenvalues from far field data

Fioralba Cakoni; David Colton; Houssem Haddar

doi:10.1016/j.crma.2010.02.003

Partial Differential Equations/Numerical Analysis

On the determination of Dirichlet or transmission eigenvalues from far field data

Presented by: Pierre-Louis Lions

Fioralba Cakoni ¹; David Colton ¹; Houssem Haddar ²

¹ Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, USA
² INRIA Saclay Ile de France & École polytechnique (CMAP), route de Saclay, 91128 Palaiseau cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 379-383.

Abstract
Résumé

We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for −Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.

Nous montrons qu'une certaine norme de l'onde de Herglotz ayant pour noyau la régularisée de Tikhonov de la solution de l'équation de champs lointains tend vers ∞ lorsque le paramètre de régularisation tend vers 0, si le nombre d'onde k appartient à un ensemble discret de valeurs. Lorsque l'objet diffractant est tel que l'onde s'annule sur sa frontière, ces valeurs sont les racines carrées des valeurs propres de Dirichlet pour −Δ. Lorsque l'objet diffractant est un milieu pénétrable non absorbant, ces valeurs coincident avec les dites valeurs propres de transmission.

Received: 2009-07-08
Accepted: 2010-02-02
Published online: 2010-03-09

DOI: 10.1016/j.crma.2010.02.003

Author's affiliations:

Fioralba Cakoni ¹; David Colton ¹; Houssem Haddar ²

¹ Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, USA
² INRIA Saclay Ile de France & École polytechnique (CMAP), route de Saclay, 91128 Palaiseau cedex, France

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     author = {Fioralba Cakoni and David Colton and Houssem Haddar},
     title = {On the determination of {Dirichlet} or transmission eigenvalues from far field data},
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Fioralba Cakoni; David Colton; Houssem Haddar. On the determination of Dirichlet or transmission eigenvalues from far field data. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 379-383. doi : 10.1016/j.crma.2010.02.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.02.003/

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Cited by Sources:

^☆ The research of F.C. and D.C. was supported in part by the U.S. Air Force Office of Scientific Research under Grant FA-9550-08-1-0138. This research was in part supported by the associate team ISIP of INRIA-UDEL.

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