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

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.

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.

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Accepté le :
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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
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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/

[1] A. Bahri; Y. Li; O. Rey 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] C. Bandle; M. Flucher Harmonic radius and concentration of energy; hyperbolic radius and Liouville's equations ΔU=eU and ΔU=U(n+2)/(n2), SIAM Rev., Volume 38 (1996) no. 2, pp. 191-238

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

[4] S. Chanillo; Y. Li Continuity of solutions of uniformly elliptic equations in R2, Manuscripta Math., Volume 77 (1992), pp. 415-433

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

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

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

[8] Z.C. Han 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] C.E. Kenig; W.M. Ni On the elliptic equation Luk+Kexp[2u]=0, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 12 (1985), pp. 191-224

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

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

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

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

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

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

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