Uniform bounds for quotients of Green functions on ${C}^{1,1}$-domains
Annales de l'Institut Fourier, Tome 32 (1982) no. 1, p. 105-117
Soient $\Delta u={\Sigma }_{i}\frac{{\partial }^{2}}{{\partial }_{{x}_{i}}^{2}}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^{2}}{\partial {x}_{i}\partial {x}_{j}}u+{\Sigma }_{i}{b}_{i}\frac{\partial }{\partial {x}_{i}}u+cu$ des opérateurs elliptiques à coefficients höldériens sur un domaine borné $\Omega \subset {\mathbf{R}}^{n}$ 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}_{\Delta }^{\Omega }\le {G}_{L}^{\Omega }\le c{G}_{\Delta }^{\Omega }\phantom{\rule{4pt}{0ex}}\text{sur}\phantom{\rule{4pt}{0ex}}\Omega ×\Omega ,$${\gamma }_{\Delta }^{\Omega }$ (resp. ${G}_{L}^{\Omega }$) étant la fonction de Green de $\Delta$ (resp. $L$) sur $\Omega$.
Let $\Delta u={\Sigma }_{i}\frac{{\partial }^{2}}{{\partial }_{{x}_{i}}^{2}}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^{2}}{\partial {x}_{i}\partial {x}_{j}}u+{\Sigma }_{i}{b}_{i}\frac{\partial }{\partial {x}_{i}}u+cu$ be elliptic operators with Hölder continuous coefficients on a bounded domain $\Omega \subset {\mathbf{R}}^{n}$ 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${c}^{-1}{G}_{\Delta }^{\Omega }\le {G}_{L}^{\Omega }\le c{G}_{\Delta }^{\Omega }\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\Omega ×\Omega ,$where ${\gamma }_{\Delta }^{\Omega }$ (resp. ${G}_{L}^{\Omega }$) are the Green functions of $\Delta$ (resp. $L$) on $\Omega$.
@article{AIF_1982__32_1_105_0,
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},
publisher = {Imprimerie Louis-Jean},
volume = {32},
number = {1},
year = {1982},
pages = {105-117},
doi = {10.5802/aif.861},
zbl = {0465.35028},
mrnumber = {84a:35063},
language = {en},
url = {http://www.numdam.org/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) no. 1, pp. 105-117. doi : 10.5802/aif.861. http://www.numdam.org/item/AIF_1982__32_1_105_0/

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