Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

By S. Armstrong, Tuomo Kuusi, Jean-Christophe Mourrat. 2015

87 citations recorded by the source corpus.

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (Ck, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities), which yields, by a new “multiscale” Poincaré inequality, quantitative estimates on the sublinearity of the corrector.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e6b8887154f3e507f970a1ad3f67bca126b0cc8

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Citations in the corpus

Quantitative stochastic homogenization and regularity theory of parabolic equations (2017, methodology): “In this paper, motivated by the desire to obtain quantitative homogenization results—in particular, explicit estimates of the homogenization error—we develop an analytic approach for parabolic equations with random coefficients based on the ideas recently introduced in [6, 5, 3, 4], which are perhap”
The additive structure of elliptic homogenization (2016, methodology): “The argument, inspired by the ideas of [4, Section 3] and our previous paper [2], relies on the connection between the gradient of the energy quantity Jk and the spatially averaged gradient and flux of its maximizers (given in (4.”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “Compared to the series [4, 6, 7], our approach has advantages both in terms of stochastic results and large-scale regularity.”
A higher-order large-scale regularity theory for random elliptic operators (2015, background): “In [1], the authors do not construct higher-order correctors and instead compare the solution to usual polynomials of degree k; this is the main reason for the restriction of the excess-decay result in [1].”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, methodology): “25 above) and originates from the ideas and philosophy of [15, 11, 12].”