Quantitative stochastic homogenization of convex integral functionals

By S. Armstrong, Charles K. Smart. 2014

242 citations recorded by the source corpus.

We present quantitative results for the homogenization of uniformly convex integral functionals with random coefficients under independence assumptions. The main result is an error estimate for the Dirichlet problem which is algebraic (but sub-optimal) in the size of the error, but optimal in stochastic integrability. As an application, we obtain quenched $C^{0,1}$ estimates for local minimizers of such energy functionals.

Subfields: Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/664ee8ff7b923de2cd7eaa9784984e54b42636e7

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Citations in the corpus

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, methodology): “s, our arguments yield estimates which are much stronger in stochastic integrability (under strong mixing assumptions, we get exponential moments rather than just pth moments). This is because, as in [4, 2], our method allows us to use subadditivity to control the stochastic ﬂuctuations. So far, the”
Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, background): “random coeﬃcients. This regularity theory in the large was ﬁrst introduced by Avellaneda & Lin [6] in the setting of periodic coeﬃcient ﬁelds, extended to the random case by Armstrong & Smart [5], and reﬁned in [21]. We refer the reader to the companion article [21] for a thorough discussion”
Boundary Estimates in Elliptic Homogenization (2015, methodology): “thanks Carlos E. Kenig for several very helpful discussions regarding this work. The author also would like to thank Scott N. Armstrong for insightful conversations and discussions regarding the work [4]. 2 ConvergenceratesinH1 In this section we establish certain results on convergence rates in H1,”
Quantitative results on the corrector equation in stochastic homogenization (2014, background): “In [2], they develop a quantitative stochastic homogenization theory for (nonlinear) convex integral functionals based on a quantification of the subadditive ergodic theorem for fields with finite range of dependence, and get suboptimal algebraic rates of convergence for the Dirichlet problem.”
Lipschitz Regularity for Elliptic Equations with Random Coefficients (2014, methodology): “near (i.e., a(p,x) = A(x)pfor a square matrix Awith positive eigenvalues), then this reduces to the requirement that Abe symmetric. More recently, Gloria, Neukamm and Otto [24], partially inspired by [3], obtained a quenched Lipschitz estimate for linear equations and systems. In particular, they gi”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, methodology): “The results in this chapter first appeared in [15] and were extended to the case of general monotone maps (that is, for a not necessarily of the form (11.”
Lipschitz Estimates in Almost‐Periodic Homogenization (2014, methodology): “Both are based on a Campanato iteration to obtain an improvement of flatness for solutions, although “flatness” in [1], and in this paper, is defined with respect to affine functions, while [11], following [2, 4], defines it with respect to correctors.”
The additive structure of elliptic homogenization (2016, background): “, renormalization) argument, which relies on the higher regularity theory developed in [4, 17].”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “which decays arbitrarily slowly, cf. Theorems 3 & 4, and obtain the optimal stochastic integrability (as opposed to the nearly-optimal stochastic integrability one would get using the approach of [5]). The deﬁnition of r∗ is based on an extended corrector (φ,σ), see Lemma 1, which allows to repr”