[Sur la détermination des fréquences propres de Dirichlet ou de transmission à partir de l'opérateur de champs lointains]
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.
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@article{CRMATH_2010__348_7-8_379_0,
author = {Cakoni, Fioralba and Colton, David and Haddar, Houssem},
title = {On the determination of {Dirichlet} or transmission eigenvalues from far field data},
journal = {Comptes Rendus. Math\'ematique},
pages = {379--383},
publisher = {Elsevier},
volume = {348},
number = {7-8},
year = {2010},
doi = {10.1016/j.crma.2010.02.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.02.003/}
}
TY - JOUR AU - Cakoni, Fioralba AU - Colton, David AU - Haddar, Houssem TI - On the determination of Dirichlet or transmission eigenvalues from far field data JO - Comptes Rendus. Mathématique PY - 2010 SP - 379 EP - 383 VL - 348 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.02.003/ DO - 10.1016/j.crma.2010.02.003 LA - en ID - CRMATH_2010__348_7-8_379_0 ER -
%0 Journal Article %A Cakoni, Fioralba %A Colton, David %A Haddar, Houssem %T On the determination of Dirichlet or transmission eigenvalues from far field data %J Comptes Rendus. Mathématique %D 2010 %P 379-383 %V 348 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.02.003/ %R 10.1016/j.crma.2010.02.003 %G en %F CRMATH_2010__348_7-8_379_0
Cakoni, Fioralba; Colton, David; Haddar, Houssem. On the determination of Dirichlet or transmission eigenvalues from far field data. Comptes Rendus. Mathématique, Tome 348 (2010) no. 7-8, pp. 379-383. doi : 10.1016/j.crma.2010.02.003. http://www.numdam.org/articles/10.1016/j.crma.2010.02.003/
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☆ 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.
