Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer
نویسندگان
چکیده
We study the behavior of the electromagnetic field in a biological cell modelized by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain our asymptotic conditions in a simple way. This paper extends to time-harmonic Maxwell equations the previous works presented in [27, 29, 28, 4]. Asymptotic transmission conditions at any order are given in Appendix 1. Key-words: Asymptotic analysis, Maxwell’s equations, differential calculus ∗ Université de Rennes1, I.R.M.A.R. & CNRS UMR 6625, Campus de Beaulieu, 35042 Rennes Cedex, France † INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 & Université de Bordeaux1, 351 cours de la Libération, 33405 Talence Cedex, France in ria -0 03 47 97 1, v er si on 3 14 S ep 2 00 9 Conditions de transmission approchées pour les équations de Maxwell en régime harmonique dans un milieu à couche mince Résumé : Nous étudions le comportement asymptotique du champ électromagnétique dans une cellule biologique plongée dans un milieu ambiant. La cellule est composée d’un cytoplasme entouré d’une fine membrane. Nous obtenons des conditions de transmission sur le bord du cytoplasme équivalentes à la couche mince. Notre approche est essentiellement géométrique et basée sur un changement de variables adéquat dans la couche mince. Quelques notions de calcul différentiel sont rappelées afin d’obtenir directement notre développement asymptotique. Par ailleurs des estimations d’erreur sont démontrées. En appendice, nous présentons le développement asymptotique à tout ordre. Mots-clés : Analyse Asymptotique, Equations de Maxwell, Calcul différentiel in ria -0 03 47 97 1, v er si on 3 14 S ep 2 00 9 Approximate Transmission Conditions 3
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