Annealed estimates on the Green functions and uncertainty quantification

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• 作者：Antoine Gloria^a ; ^b ; ^{antoine.gloria@ulb.ac.be" class="auth_mail" title="E-mail the corresponding author} ; Daniel Marahrens^c ; ^{daniel.marahrens@mis.mpg.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J08 ; 35J15 ; 60K37 ; 60H25 ; 35B65
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：September-October 2016
• 年：2016
• 卷：33
• 期：5
• 页码：1153-1197
• 全文大小：641 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0294144915000414&_mathId=si1.gif&_user=111111111&_pii=S0294144915000414&_rdoc=1&_issn=02941449&md5=53e292d8b3a61e4a8fab61bd60ed7413" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$. As another application of these decay estimates we derive optimal estimates on the fluctuations of solutions of linear elliptic PDEs with “noisy” diffusion coefficients.