Green's function for elliptic systems: Moment bounds

Bella, Peter; Giunti, Arianna

doi:10.3934/nhm.2018007

Cited by 11 publications

(10 citation statements)

References 12 publications

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“…In order to pass from Theorem 2 to Corollary 3, we need the following statement on families (rather ensembles) of a-harmonic functions, which is of independent interest and motivated by [4].…”

Section: Resultsmentioning

confidence: 99%

Quantitative stochastic homogenization: Local control of homogenization error through corrector

Bella¹,

Giunti²,

Otto³

et al. 2017

Mathematics and Materials

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This note addresses the homogenization error for linear elliptic equations in divergenceform with random stationary coefficients. The homogenization error is measured by comparing the quenched Green's function to the Green's function belonging to the homogenized coefficients, more precisely, by the (relative) spatial decay rate of the difference of their second mixed derivatives. The contribution of this note is purely deterministic: It uses the expanded notion of corrector, namely the couple of scalar and vector potentials (φ, σ), and shows that the rate of sublinear growth of (φ, σ) at the points of interest translates one-to-one into the decay rate.

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Section: Resultsmentioning

confidence: 99%

Quantitative stochastic homogenization: Local control of homogenization error through corrector

Bella¹,

Giunti²,

Otto³

et al. 2017

Mathematics and Materials

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“…Finally, we turn to the proof of (3.11) in the critical case β = d. In order to obtain the optimal power of the logarithm, we rather use the Green's representation formula for ∇Φ 0 and appeal to annealed bounds on the Green's function [29,14,3,16,6] in the form…”

Section: Convergence Of the Covariance Structurementioning

confidence: 99%

Scaling limit of the homogenization commutator for Gaussian coefficient fields

Duerinckx¹,

Fischer²,

Gloria³

2019

Preprint

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Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution-operator displays fluctuations around itsexpectation. The recently-developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension d ≥ 1 previous results so far limited to dimension d = 1. Contents 1. Introduction 1 2. Preliminary 5 3. Convergence of the covariance structure 9 4. (Non-)Degeneracy of the limiting covariance 17 5. Asymptotic normality 23 Acknowledgments 30 References 30

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“…On a technical level, we leverage and extend techniques developed in the wider context of homogenization theory for controlling Green's functions. This is a wide field; see [2,5,6,7,9,11,12,13,15,21,22,30,31,41,46,47] and references therein. Specifically, we draw on upper bounds proved by Marahrens-Otto [46,47] and add some new results on lower bounds and asymptotics of Green's functions and their derivatives.…”

Section: Introductionmentioning

confidence: 99%

Optimal Hardy weights on the Euclidean lattice

Keller¹,

Lemm²

2021

Preprint

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We investigate the large-distance asymptotics of optimal Hardy weights on Z d , d ≥ 3, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiarHere we prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on Z d . Namely, (1) annular averages have inverse-square scaling in general, (2), for ergodic coefficients there is an almost-sure pointwise inverse-square upper bound, and (3), for weakly random i.i.d. coefficients there is a probabilistic inverse-square lower bound. The results have continuum analogs and also yield |x| −4 -scaling for Rellich weights on Z d . The technique relies on Green's function estimates rooted in homogenization theory. Along the way, we identify for the first time an asymptotic expansion of the averaged Green's function and of its derivatives up to order d+1 at small ellipticity contrast. This leverages a recent perturbative approach to homogenization theory developed by Bourgain. The result implies rather universal asymptotics of the random Green's function via concentration bounds of Marahrens-Otto.

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Green's function for elliptic systems: Moment bounds

Cited by 11 publications

References 12 publications

Quantitative stochastic homogenization: Local control of homogenization error through corrector

Quantitative stochastic homogenization: Local control of homogenization error through corrector

Scaling limit of the homogenization commutator for Gaussian coefficient fields

Optimal Hardy weights on the Euclidean lattice

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