Uniform bounds for quotients of Green functions on $C^{1,1}$-domains

Hueber, H.; Sieveking, M.

Uniform bounds for quotients of Green functions on

C^{1, 1}

-domains

Hueber, H. ; Sieveking, M.

Annales de l'Institut Fourier, Tome 32 (1982), p. 105-117 / Harvested from Numdam

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Résumé
Abstract

Soient $Δ u = Σ_{i} \frac{\partial^{2}}{\partial_{x_{i}}^{2}}$ , $L u = Σ_{i, j} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} u + Σ_{i} b_{i} \frac{\partial}{\partial x_{i}} u + c u$ des opérateurs elliptiques à coefficients höldériens sur un domaine borné $Ω \subset R^{n}$ de classe $C^{1, 1}$ . Il existe une constante $c > 0$ ne dépendant que des normes de Hölder des coefficients de $L$ et de sa constante d’ellipticité telle que

c^{- 1} G_{Δ}^{Ω} \leq G_{L}^{Ω} \leq c G_{Δ}^{Ω} sur Ω \times Ω,

$γ_{Δ}^{Ω}$ (resp. $G_{L}^{Ω}$ ) étant la fonction de Green de $Δ$ (resp. $L$ ) sur $Ω$ .

Let $Δ u = Σ_{i} \frac{\partial^{2}}{\partial_{x_{i}}^{2}}$ , $L u = Σ_{i, j} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} u + Σ_{i} b_{i} \frac{\partial}{\partial x_{i}} u + c u$ be elliptic operators with Hölder continuous coefficients on a bounded domain $Ω \subset R^{n}$ of class $C^{1, 1}$ . There is a constant $c > 0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that

c^{- 1} G_{Δ}^{Ω} \leq G_{L}^{Ω} \leq c G_{Δ}^{Ω} on Ω \times Ω,

where $γ_{Δ}^{Ω}$ (resp. $G_{L}^{Ω}$ ) are the Green functions of $Δ$ (resp. $L$ ) on $Ω$ .

Publié le : 1982-01-01

MR 84a:35063 | Zbl 0465.35028

DOI : https://doi.org/10.5802/aif.861

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     author = {Hueber, H. and Sieveking, M.},
     title = {Uniform bounds for quotients of Green functions on $C^{1,1}$-domains},
     journal = {Annales de l'Institut Fourier},
     volume = {32},
     year = {1982},
     pages = {105-117},
     doi = {10.5802/aif.861},
     mrnumber = {84a:35063},
     zbl = {0465.35028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1982__32_1_105_0}
}

Hueber, H.; Sieveking, M. Uniform bounds for quotients of Green functions on $C^{1,1}$-domains. Annales de l'Institut Fourier, Tome 32 (1982) pp. 105-117. doi : 10.5802/aif.861. http://gdmltest.u-ga.fr/item/AIF_1982__32_1_105_0/

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