[Sur un guide d'onde avec un nombre infini de petites fentes]
On considère un guide d'onde modélisé par le Laplacien dans une bande horizontale, avec des conditions de Dirichlet sur le bord supérieur et des conditions du type Dirichlet et Neumann qui alternent périodiquement sur le bord inférieur. La période est considérée petite et on étudie le problème de l'homogénéisation : on démontre la convergence en norme de la résolvante vers la résolvante du Laplacien avec des conditions de Neumann sur tout le bord inférieur et on obtient des estimations du taux de convergence. Ensuite on donne les deux premiers termes du développement asymptotique des valeurs propres de l'opérateur perturbé, ainsi que le développement asymptotique complet du bas de son spectre.
We consider a waveguide modeled by the Laplacian in a straight planar strip with the Dirichlet condition on the upper boundary, while on the lower one we impose periodically alternating boundary conditions with a small period. We study the case when the homogenization leads us to the Neumann boundary condition on the lower boundary. We establish the uniform resolvent convergence and provide the estimates for the rate of convergence. We construct the two-terms asymptotics for the first band functions of the perturbed operator and also the complete two-parametric asymptotic expansion for the bottom of its spectrum.
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@article{CRMATH_2011__349_1-2_53_0,
author = {Denis Borisov and Renata Bunoiu and Giuseppe Cardone},
title = {On a waveguide with an infinite number of small windows},
journal = {Comptes Rendus. Math\'ematique},
pages = {53--56},
publisher = {Elsevier},
volume = {349},
number = {1-2},
year = {2011},
doi = {10.1016/j.crma.2010.11.029},
language = {en},
}
Denis Borisov; Renata Bunoiu; Giuseppe Cardone. On a waveguide with an infinite number of small windows. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 53-56. doi : 10.1016/j.crma.2010.11.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.029/
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