Corrector Estimates for Elliptic Systems with Random Periodic Coefficients

Bella, Peter; Otto, Félix

doi:10.1137/15m1037147

Cited by 15 publications

(21 citation statements)

References 24 publications

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“…We conclude this subsection by mentioning two more recent related works: both Ben-Artzi, Marahrens and Neukamm [8] and Bella and Otto [7] obtained moment bounds on the gradient of the approximate correctors for linear elliptic systems using concentration inequalities, and Lamacz, Neukamm and Otto [30] obtained such estimates for a percolation model using similar methods.…”

Section: Motivation and Previous Workmentioning

confidence: 87%

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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Section: Motivation and Previous Workmentioning

confidence: 87%

Lipschitz Regularity for Elliptic Equations with Random Coefficients

Armstrong

Mourrat

2015

Arch Rational Mech Anal

111

202

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“…In particular, for 0 ≤ β < 1 − 2 d , this implies the existence and uniqueness of a stationary extended corrector (φ, σ) with finite second moment that solves (6)- (8).…”

Section: 4mentioning

confidence: 90%

“…We note that according to (8) and (30), σ−σ t satisfies − (σ−σ t ) = ∇×(q−q t )+ 1 t σ t . On B R , we split σ − σ t = u + w according to…”

Section: 2mentioning

confidence: 99%

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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“…We remark that it is completely arbitrary whetherũ h orṽ h takes care of the extra term´| x|=R ν k ∂ i v ′ h ∂ j u ′ C sym ijk . Existence and uniqueness ofũ h andṽ h , as defined by (13), (15), and (14), (16), are established in Lemma 9 and Lemma 10, respectively. Remark 1.…”

Section: The Main Resultsmentioning

confidence: 97%

“…whereũ h is associated to u h according to (13), (15) andṽ h to v h according to (14), (16) (see Lemma 9 and Lemma 10 for their construction).…”

Section: Deterministic Resultsmentioning

confidence: 99%

Effective multipoles in random media

Bella

Giunti

Otto

2020

Communications in Partial Differential Equations

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In a homogeneous medium, the far-field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole -but not the octupole -can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest.Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero-and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green's formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials.

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Corrector Estimates for Elliptic Systems with Random Periodic Coefficients

Cited by 15 publications

References 24 publications

Lipschitz Regularity for Elliptic Equations with Random Coefficients

Lipschitz Regularity for Elliptic Equations with Random Coefficients

A Regularity Theory for Random Elliptic Operators

Effective multipoles in random media

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