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

Conlon, Joseph G.; Giunti, Arianna; Otto, Félix

doi:10.1007/s00526-017-1255-0

Cited by 12 publications

(38 citation statements)

References 30 publications

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“…Nevertheless, as recently shown in [7] by Conlon, Otto, and the second author, this is not a generic behavior. More precisely, in [7] they show that for any uniformly elliptic coefficient field A the Green's function G = G(A; x, y) exists at almost every point y ∈ R d , provided the dimension d ≥ 3. Therefore, in the case d ≥ 3, we will assume that the Green's function G(A; ·, y) ∈ L 1 loc (R d ) exists, at least in the almost everywhere sense (i.e., for a.e.…”

Section: Introductionmentioning

confidence: 53%

Green's function for elliptic systems: Moment bounds

Bella

Giunti

2018

Networks &Amp; Heterogeneous Media

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We study estimates of the Green's function in R d with d ≥ 2, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case d ≥ 3, we obtain estimates on the Green's function, its gradient, and the second mixed derivatives which scale optimally in space, in terms of the "minimal radius" r * introduced in [Gloria, Neukamm, and Otto: A regularity theory for random elliptic operators; ArXiv e-prints (2014)]. As an application, our result implies optimal stochastic Gaussian bounds in the realm of homogenization of equations with random coefficient fields with finite range of dependence. In two dimensions, where in general the Green's function does not exist, we construct its gradient and show the corresponding estimates on the gradient and mixed second derivatives. Since we do not use any scalar methods in the argument, the result holds in the case of uniformly elliptic systems as well.

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Section: Introductionmentioning

confidence: 53%

Green's function for elliptic systems: Moment bounds

Bella

Giunti

2018

Networks &Amp; Heterogeneous Media

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“…In view of (19) and (20) the flux a(e i + ∇φ i ) is divergence-free and of expectation a h e i . It is thus natural to consider, next to the correction φ i of the (scalar) potential of the closed 1-form e i + ∇φ i , also the correction of the (vector) potential of the closed (d − 1)-form a(e i + ∇φ i ).…”

Section: First and Second-order Correctors In Stochastic Homogenizationmentioning

confidence: 99%

“…• The random fields φ, σ, ∇ψ, ∇Ψ, and r * are stationary; ∇φ, φ, ∇σ, σ, ∇ψ, and ∇Ψ have finite second moments, so that in particular a h and C are well-defined through (20) and (25). We impose the normalization…”

Section: First and Second-order Correctors In Stochastic Homogenizationmentioning

confidence: 99%

“…Let · be a stationary ensemble satisfying an LSI. Then the coefficient a h ∈ R d×d defined in (20) satisfies…”

Section: Corollarymentioning

confidence: 99%

“…Also this paper allows for systems which is of interest in particular because of the system of linear elasticity. Incidentally, equipped with [20,Corollary 2], also [39] extends to systems. The subsequent work of Gloria, Neukamm and the third author [28] refined this Campanato-approach by working with the "flux corrector" σ (or vector potential of the flux), see (21), which plays a crucial technical role in the present paper, and which is also known from periodic homogenization [47, p.27].…”

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confidence: 99%

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Effective multipoles in random media

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

Cited by 12 publications

References 30 publications

Green's function for elliptic systems: Moment bounds

Green's function for elliptic systems: Moment bounds

Effective multipoles in random media

A Regularity Theory for Random Elliptic Operators

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