Comptes Rendus
no. 7-8
Partial Differential Equations/Numerical Analysis
On the determination of Dirichlet or transmission eigenvalues from far field data
[Sur la détermination des fréquences propres de Dirichlet ou de transmission à partir de l'opérateur de champs lointains]

Présenté par : Pierre-Louis Lions

Fioralba Cakoni1 ; David Colton1 ; Houssem Haddar2
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.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.02.003
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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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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[4] F. Cakoni, D. Colton, H. Haddar, The interior transmission problem for region with cavities, SIAM Jour. Math. Anal., | DOI

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[6] D. Colton Partial Differential Equations: An Introduction, Dover, New York, 2004

[7] A. Kirsch Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, Volume 15 (1999), pp. 413-429

[8] J. Lions; E. Magenese Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, NY, 1972

[9] L. Päivärinta; J. Sylvester Transmission eigenvalues, SIAM J. Math. Anal., Volume 40 (2008), pp. 738-753

[10] B.P. Rynne; B.D. Sleeman The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., Volume 22 (1992), pp. 1755-1762

Cité par 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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