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We prove Lipschitz bounds for linear elliptic equations in divergence form whose measurable coefficients are random stationary and satisfy a logarithmic Sobolev inequality, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. This improves the celebrated De Giorgi–Nash–Moser theory in the large (that is, away from the singularity) for this class of coefficients. This regularity result is obtained as a corollary of optimal decay estimates on the derivative and mixed second derivative of the elliptic Green functions on . As another application of these decay estimates we derive optimal estimates on the fluctuations of solutions of linear elliptic PDEs with “noisy” diffusion coefficients.
Keywords: Green's functions, Elliptic equations, Annealed estimates, Uncertainty quantification, Regularity theory
@article{AIHPC_2016__33_5_1153_0,
author = {Gloria, Antoine and Marahrens, Daniel},
title = {Annealed estimates on the {Green} functions and uncertainty quantification},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1153--1197},
publisher = {Elsevier},
volume = {33},
number = {5},
year = {2016},
doi = {10.1016/j.anihpc.2015.04.001},
mrnumber = {3542610},
zbl = {1351.35027},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.04.001/}
}
TY - JOUR AU - Gloria, Antoine AU - Marahrens, Daniel TI - Annealed estimates on the Green functions and uncertainty quantification JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 1153 EP - 1197 VL - 33 IS - 5 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.04.001/ DO - 10.1016/j.anihpc.2015.04.001 LA - en ID - AIHPC_2016__33_5_1153_0 ER -
%0 Journal Article %A Gloria, Antoine %A Marahrens, Daniel %T Annealed estimates on the Green functions and uncertainty quantification %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 1153-1197 %V 33 %N 5 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.04.001/ %R 10.1016/j.anihpc.2015.04.001 %G en %F AIHPC_2016__33_5_1153_0
Gloria, Antoine; Marahrens, Daniel. Annealed estimates on the Green functions and uncertainty quantification. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 5, pp. 1153-1197. doi: 10.1016/j.anihpc.2015.04.001
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