Quantitative results on the corrector equation in stochastic homogenization

By Antoine Gloria, Felix Otto. 2014

129 citations recorded by the source corpus.

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/88058b06562c09b96af82310ed943e45d88aac37

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In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, background): “ts are due Yurinskii [24]. Developing a quantitative theory of stochastic homogenization for linear elliptic equations has received a lot of attention since the groundbreaking work of Gloria and Otto [13, 14, 15] and Gloria, Neukamm and Otto [10, 11], who proved an array of optimal estimates under c”
Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, methodology): “; • Third, we establish a quantitative two-scale expansion in the spirit of [20], for which the scaling of the error now depends on πand d, cf. Theorem 3. These results extend our previous results in [25] in various respects: They apply to systems with correlated coeﬃcients, and yield stretched expo”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “1, but with much weaker stochastic integrability, were proved several years earlier in [65, 66, 62, 67] for a certain class of random environments”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “In contrast to our framework based on nonlinear mixing conditions (as developed in [41, 43, 38] and the present paper), Armstrong and Smart quantify ergodicity in terms of linear mixing conditions.”
Quantitative stochastic homogenization of convex integral functionals (2014, background): “ria Neukamm and Otto [14, 13], who proved optimal bounds for the error in homogenization. These results were proved for discrete elliptic equations, but have been extended to the continuum setting in [17]. See also Mourrat [20, 21], Marahrens and Otto [19], Conlon and Spencer [7] as well as earlier ”
Lipschitz Regularity for Elliptic Equations with Random Coefficients (2014, methodology): “ut that good estimates of the latter, combined with appropriate concentration inequalities, yield optimal quantitative bounds in homogenization. This was shown for linear equations by Gloria and Otto [25, 26, 27] and Gloria, Neukamm and Otto [22, 23]. The work Date: June 30, 2015. 2010 Mathematics S”
The additive structure of elliptic homogenization (2016, methodology): “lev inequalities, which provide a way to quantitatively measure the dependence of the solutions on the coeﬃcients. This approach has been developed extensively by Gloria, Otto and their collaborators [15, 16, 18, 12, 11, 21], who proved optimal quantitative bounds on the scaling of the ﬁrst-order co”