On Homogenization Of Elliptic Equations With Random Coefficients

Conlon, Joseph G.; Naddaf, Ali

doi:10.1214/ejp.v5-65

Cited by 42 publications

(109 citation statements)

References 4 publications

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“…As introduced in Sect. 2, for a given a ∈ , we treat the Green function for a domain D as an object G D (a; ·, ·) in two space variables (x, y) ∈ R d × R d , which satisfies for almost every singularity point y ∈ R d the equation (8). It is with this generalised definition of Green function that we manage to prove its existence and uniqueness (cf.…”

Section: Main Results and Remarksmentioning

confidence: 99%

“…In the last section we present an alternative partial proof for Corollary 2 which makes use of the Fourier techniques developed in [8] and relies on a representation formula for the Fourier transform of the Green function. Finally, in the Appendix we give a self-contained proof of all the auxiliary results which are used in the arguments.…”

Section: Main Results and Remarksmentioning

confidence: 99%

“…It is an interesting exercise to compare the methodology used in the present paper with the methodology used by the first author and Naddaf [8] to prove in the scalar case pointwise estimates on the averaged Green function. and their derivatives.…”

Section: Estimates (1) Immediately Imply Optimal Decay Bounds For Thementioning

confidence: 99%

“…and their derivatives. While in this work we infer the stochastic bounds on the Green function directly from the deterministic existence result for G(a; ·, ·), in [8] a major part is played by the Fourier representation of the averaged Green function, which is a generalisation of the Fourier representation of the Green function for an elliptic PDE with constant coefficients. Naddaf and the first author then obtain Fourier space estimates strong enough to imply the pointwise estimates on averaged Green functions.…”

Section: Estimates (1) Immediately Imply Optimal Decay Bounds For Thementioning

confidence: 99%

“…Naddaf and the first author then obtain Fourier space estimates strong enough to imply the pointwise estimates on averaged Green functions. We remark that the method in [8] does not make use of the scalar structure and therefore may be applied also to obtain a-priori estimates in the vectorial case. In the last section of this paper we indeed summarise how our main estimate in the proof of (1) can be obtained using this Fourier method.…”

Section: Estimates (1) Immediately Imply Optimal Decay Bounds For Thementioning

confidence: 99%

See 4 more Smart Citations

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

2017

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This paper is divided into two parts: In the main deterministic part, we prove that for an open domain D ⊂ R d with d ≥ 2, for every (measurable) uniformly elliptic tensor field a and for almost every point y ∈ D, there exists a unique Green's function centred in y associated to the vectorial operator −∇ · a∇ in D. This result implies the existence of the fundamental solution for elliptic systems when d > 2, i.e. the Green function for −∇ · a∇ in R d . In the second part, we introduce a shift-invariant ensemble · over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for |G(·; x, y)| , |∇ x G(·; x, y)| and |∇ x ∇ y G(·; x, y)| . These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case. Mathematics Subject Classification

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Section: Main Results and Remarksmentioning

confidence: 99%

Section: Main Results and Remarksmentioning

confidence: 99%

Section: Estimates (1) Immediately Imply Optimal Decay Bounds For Thementioning

confidence: 99%

Section: Estimates (1) Immediately Imply Optimal Decay Bounds For Thementioning

confidence: 99%

Section: Estimates (1) Immediately Imply Optimal Decay Bounds For Thementioning

confidence: 99%

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Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

2017

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A Regularity Theory for Random Elliptic Operators

2020

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The qualitative theory of stochastic homogenization of uniformly elliptic linear (but possibly non-symmetric) systems in divergence form is well-understood. Quantitative results on the speed of convergence and on the error in the representative volume method, like those recently obtained by the authors for scalar equations, require a type of stochastic regularity theory for the corrector (e.g., higher moment bounds). One of the main insights of the very recent work of Armstrong and Smart is that one should separate these error estimates, which require strong mixing conditions in order to yield the best rates possible, from the (large scale) regularity theorỳ a la Avellaneda & Lin for a-harmonic functions that should hold under milder mixing conditions. In this paper, we establish an intrinsinc C 1,1 -version of the improved regularity theory for non-symmetric random systems, that is qualitative in the sense it yields a new Liouville theorem under mere ergodicity, and that is quantifiable in the sense that it holds under a condition that has high stochastic integrabiliity provided the coefficients satisfy quantitative mixing assumptions. We introduce such a class of quantitative mixing condition, that allows for arbitrarily slow-decaying correlations, and under which we derive a new family of optimal estimates in stochastic homogenization.

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Lipschitz Regularity for Elliptic Equations with Random Coefficients

Armstrong

Mourrat

2015

Arch Rational Mech Anal

111

202

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Abstract. We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L ∞ -type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched L 2 estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

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On Homogenization Of Elliptic Equations With Random Coefficients

Cited by 42 publications

References 4 publications

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

A Regularity Theory for Random Elliptic Operators

Lipschitz Regularity for Elliptic Equations with Random Coefficients

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