Comptes Rendus
Partial Differential Equations
Expansion of the Green's function for divergence form operators
Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 891-896.

We consider the fundamental solution Ga of the operator Δa=1a(x)div(a(x)) on a bounded smooth domain ΩRn (n2), associated to the Dirichlet boundary condition, where a is a positive smooth function on Ω¯. In this short Note, we give a precise description of the function Ga(x,y). In particular, we define in a unique way its continuous part Ha(x,y) and we prove that the corresponding Robin's function Ra(x)=Ha(x,x) belongs to C(Ω), although HaC1(Ω×Ω) in general.

On considère la solution fondamentale Ga de l'opérateur Δa=1a(x)div(a(x)), sur un domaine borné régulier ΩRn (n2) avec les conditions de Dirichlet au bord, ici a est une fonction régulière et strictement positive sur Ω¯. Dans cette Note, on donne une description précise de la fonction Ga(x,y). On définit notamment Ha(x,y), la partie continue de Ga et on montre que la fonction de Robin correspondante Ra(x)=Ha(x,x) est dans C(Ω), sachant que HaC1(Ω×Ω) en général.

Published online:
DOI: 10.1016/j.crma.2010.06.024

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
     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},
AU  - Saïma Khenissy
AU  - Yomna Rébaï
AU  - Dong Ye
TI  - Expansion of the Green's function for divergence form operators
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 891
EP  - 896
VL  - 348
IS  - 15-16
PB  - Elsevier
DO  - 10.1016/j.crma.2010.06.024
LA  - en
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%0 Journal Article
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%A Yomna Rébaï
%A Dong Ye
%T Expansion of the Green's function for divergence form operators
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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.

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