Expansion of the Green's function for divergence form operators

Khenissy, Saïma; Rébaï, Yomna; Ye, Dong

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

Khenissy, Saïma ¹ ; Rébaï, Yomna ² ; Ye, Dong 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, Tome 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 :

Khenissy, Saïma ¹ ; Rébaï, Yomna ² ; Ye, Dong ³

@article{CRMATH_2010__348_15-16_891_0,
     author = {Khenissy, Sa{\"\i}ma and R\'eba{\"\i}, Yomna and Ye, Dong},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2010.06.024/}
}

TY  - JOUR
AU  - Khenissy, Saïma
AU  - Rébaï, Yomna
AU  - Ye, Dong
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
UR  - http://www.numdam.org/articles/10.1016/j.crma.2010.06.024/
DO  - 10.1016/j.crma.2010.06.024
LA  - en
ID  - CRMATH_2010__348_15-16_891_0
ER  -

%0 Journal Article
%A Khenissy, Saïma
%A Rébaï, Yomna
%A Ye, Dong
%T Expansion of the Green's function for divergence form operators
%J Comptes Rendus. Mathématique
%D 2010
%P 891-896
%V 348
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2010.06.024/
%R 10.1016/j.crma.2010.06.024
%G en
%F CRMATH_2010__348_15-16_891_0

Khenissy, Saïma; Rébaï, Yomna; Ye, Dong. Expansion of the Green's function for divergence form operators. Comptes Rendus. Mathématique, Tome 348 (2010) no. 15-16, pp. 891-896. doi : 10.1016/j.crma.2010.06.024. http://www.numdam.org/articles/10.1016/j.crma.2010.06.024/

Bibliographie
Cité par

[1] Bahri, A.; Li, Y.; Rey, O. On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, Volume 3 (1995), pp. 67-93

[2] Bandle, C.; Flucher, M. Harmonic radius and concentration of energy; hyperbolic radius and Liouville's equations $Δ U = e^{U}$ and $Δ U = U^{(n + 2) / (n - 2)}$ , SIAM Rev., Volume 38 (1996) no. 2, pp. 191-238

[3] Bartolucci, D.; Orsina, L. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates, Commun. Pure Appl. Anal., Volume 4 (2005) no. 3, pp. 499-522

[4] Chanillo, S.; Li, Y. Continuity of solutions of uniformly elliptic equations in $R^{2}$ , Manuscripta Math., Volume 77 (1992), pp. 415-433

[5] Del Pino, M.; Felmer, P.L. Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., Volume 149 (1997) no. 1, pp. 245-265

[6] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Springer, 2001

[7] Grüter, M.; Widman, K.O. The Green function for uniformly elliptic equations, Manuscripta Math., Volume 37 (1982) no. 3, pp. 303-342

[8] Han, Z.C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Anal. Inst. Poincaré Anal. Nonlin., Volume 8 (1991), pp. 159-174

[9] Kenig, C.E.; Ni, W.M. On the elliptic equation $L u - k + K \exp [2 u] = 0$ , Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 12 (1985), pp. 191-224

[10] Littman, W.; Stampacchia, G.; Weinberger, H.F. Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, Volume 17 (1963), pp. 43-77

[11] Maz'ya, V.; McOwen, R. On the fundamental solution of an elliptic equation in nondivergence form, 2008 | arXiv

[12] Nagasaki, K.; Suzuki, T. Asymptotic analysis for two-dimensional elliptic eigenvalue problem with exponentially dominated nonlinearities, Asymptot. Anal., Volume 3 (1990), pp. 173-188

[13] Rébaï, Y. Study of the singular Yamabe problem in some bounded domain of $R^{n}$ , Adv. Nonlinear Stud., Volume 6 (2006), pp. 437-460

[14] Rey, O. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., Volume 89 (1990) no. 1, pp. 1-52

[15] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984), pp. 479-495

[16] Wei, J.; Ye, D.; Zhou, F. Bubbling solutions for an anisotropic Emden–Fowler equation, Calc. Var. Partial Differential Equations, Volume 28 (2007), pp. 217-247

Cité par Sources :