Expansion of the Green's function for divergence form operators

Saïma Khenissy; Yomna Rébaï; Dong Ye

doi:10.1016/j.crma.2010.06.024

Partial Differential Equations

Expansion of the Green's function for divergence form operators
[Expansion de la fonction de Green pour les opérateurs de type divergence]

Présenté par : Haïm Brezis

Saïma Khenissy ¹ ; Yomna Rébaï ² ; Dong Ye ³

¹ Département de mathématiques appliquées, institut supérieur d'informatique, 2037 Ariana, Tunisia
² Département de mathématiques, faculté des sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisia
³ LMAM, UMR 7122, Université de Metz, 57045 Metz, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 891-896.

Résumé
Abstract

On considère la solution fondamentale $G_{a}$ de l'opérateur $- Δ_{a} = - \frac{1}{a (x)} div (a (x) \nabla \cdot)$ , sur un domaine borné régulier $Ω \subset R^{n}$ ( $n ⩾ 2$ ) avec les conditions de Dirichlet au bord, ici a est une fonction régulière et strictement positive sur $\bar{Ω}$ . Dans cette Note, on donne une description précise de la fonction $G_{a} (x, y)$ . On définit notamment $H_{a} (x, y)$ , la partie continue de $G_{a}$ et on montre que la fonction de Robin correspondante $R_{a} (x) = H_{a} (x, x)$ est dans $C^{\infty} (Ω)$ , sachant que $H_{a} \notin C^{1} (Ω \times Ω)$ en général.

We consider the fundamental solution $G_{a}$ of the operator $- Δ_{a} = - \frac{1}{a (x)} div (a (x) \nabla \cdot)$ on a bounded smooth domain $Ω \subset R^{n}$ ( $n ⩾ 2$ ), associated to the Dirichlet boundary condition, where a is a positive smooth function on $\bar{Ω}$ . In this short Note, we give a precise description of the function $G_{a} (x, y)$ . In particular, we define in a unique way its continuous part $H_{a} (x, y)$ and we prove that the corresponding Robin's function $R_{a} (x) = H_{a} (x, x)$ belongs to $C^{\infty} (Ω)$ , although $H_{a} \notin C^{1} (Ω \times Ω)$ in general.

Reçu le : 2010-04-19
Accepté le : 2010-06-29
Publié le : 2010-07-21

DOI : 10.1016/j.crma.2010.06.024

Affiliations des auteurs :

Saïma Khenissy ¹ ; Yomna Rébaï ² ; Dong Ye ³

¹ Département de mathématiques appliquées, institut supérieur d'informatique, 2037 Ariana, Tunisia
² Département de mathématiques, faculté des sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisia
³ LMAM, UMR 7122, Université de Metz, 57045 Metz, France

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     author = {Sa{\"\i}ma Khenissy and Yomna R\'eba{\"\i} and Dong Ye},
     title = {Expansion of the {Green's} function for divergence form operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {891--896},
     publisher = {Elsevier},
     volume = {348},
     number = {15-16},
     year = {2010},
     doi = {10.1016/j.crma.2010.06.024},
     language = {en},
}

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Saïma Khenissy; Yomna Rébaï; Dong Ye. Expansion of the Green's function for divergence form operators. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 891-896. doi : 10.1016/j.crma.2010.06.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.024/

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