Annealed estimates on the Green functions and uncertainty quantification

Abstract : We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on $\mathbb{R}^d$ for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality and for periodic coefficients, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. As a main application we obtain optimal estimates on the fluctuations of solutions of linear elliptic PDEs with "noisy" diffusion coefficients, an uncertainty quantification result. As a direct corollary of the decay estimates we also prove that for these classes of coefficients the H\"older exponent of the celebrated De Giorgi-Nash-Moser theory can be taken arbitrarily close to 1 in the large (that is, away from the singularity).
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https://hal.archives-ouvertes.fr/hal-01093386
Contributor : Antoine Gloria <>
Submitted on : Wednesday, December 10, 2014 - 3:44:48 PM
Last modification on : Monday, August 20, 2018 - 9:46:01 AM

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Antoine Gloria, Daniel Marahrens. Annealed estimates on the Green functions and uncertainty quantification. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2016, 33 (6), pp.1153--1197. ⟨10.1016/j.anihpc.2015.04.001⟩. ⟨hal-01093386⟩

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