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Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems

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Pith's one-line read Optimal homogenization rates for nonlinear random elliptic PDEs are O(ε) in d≥3, matching linear theory.

desk verdict A strong and largely explicit proof of the first optimal-order rates in nonlinear stochastic homogenization, but the main theorems lean on two key lemmas deferred to a forthcoming paper, so the proof is conditional until those appear. read the letter →

arxiv 1908.02273 v2 pith:S7W3YAEZ submitted 2019-08-06 math.AP cs.NAmath.NAmath.PR

Julian Fischer , Stefan Neukamm This is my paper · ORCID
classification math.APcs.NAmath.NAmath.PR MSC 35B2735J6060H2535R60
keywords stochastichomogenizationmonotonenonlinearellipticPDEoptimalconvergenceratespectralgapinequalitycorrectorestimatesrepresentativevolumerandomcoefficientfieldtwo-scaleexpansion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that solutions of random nonlinear uniformly elliptic equations with monotone nonlinearity converge to the homogenized solution at the optimal rate: order ε in d≥3, and order ε|log ε|^{1/2} in d=2 with a massive term. These are the same rates known for linear elliptic equations, which are a special case, so the rates cannot be improved in general. Previous nonlinear stochastic homogenization results only gave a small algebraic rate ε^δ. The paper also proves optimal-order error estimates for approximating the homogenized operator by representative volumes, with rate (L/ε)^{-d/2}. If correct, this is the first optimal-order quantitative homogenization result for any nonlinear stochastic homogenization problem.

What carries the argument

The central objects are the homogenization corrector φ_ξ (a field correcting the macroscopic gradient ξ to the microscopic gradient ξ + ∇φ_ξ) and the flux corrector σ_ξ (a d−1-form solving ∇·σ_ξ = A(ω_ε, ξ + ∇φ_ξ) − A_hom(ξ)). Alongside these stand their localized versions φ_ξ^T, σ_ξ^T with a massive term 1/T that provides exponential localization, and the correctors for the linearized PDE around ξ + ∇φ_ξ, denoted φ_{ξ,Ξ}^T and σ_{ξ,Ξ}^T. The key quantization mechanism is a spectral gap inequality on the random field ω_ε: for every random variable F(ω_ε), the variance is bounded by ε^d times the $L^{2}$ norm of a spatial average of the sensitivity ∂F/∂ω_ε. Feeding corrector sensitivities into this inequality yields optimal fluctuation bounds, and a minimal-radius construction with a hole-filling estimate turns these into L^p corrector bounds. The optimal rates emerge from combining corrector bounds with a piecewise-affine two-scale expansion of u_hom.

What would settle it

Construct a stationary random field ω_ε for which the spectral gap inequality (P2) holds but the corrector φ_ξ fails to satisfy the bound E[|∇φ_ξ|^2]^{1/2} ≲ |ξ| ε in d=3; then the paper's central homogenization-rate theorem would be false. Concretely, one could numerically compute the homogenization error for a scalar monotone equation with A(ω, ξ) = (1 + η(ω_ε(x))) ξ for a smooth Gaussian field with covariance decaying like (1 + |x−y|/ε)^{-(d+κ)}, κ>0, and check whether the error is indeed O(ε) in $L^{{2d/(d−2)}}$. A slower rate, such as ε|log ε|^{1/2}, would contradict the theorem.

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Extended reading notes

Core claim

The central claim is that for a random monotone operator A(ω_ε(x), ξ) with stationary law (P1) and spectral-gap decorrelation on scale ε (P2), and under a small-scale $C^{{1,α}}$ regularity condition (R), the solution u_ε of the random PDE and the solution u_hom of the homogenized PDE satisfy ||u_ε − u_hom||_{$L^{{2d/(d−2)}}$(R^d)} ≤ C Ĉ(∇u_hom) ε for d≥3, and ||u_ε − u_hom||_{$L^{2}$(R^d)} ≤ C Ĉ(∇u_hom) ε |log ε|^{1/2} for d=2, with C a random constant having bounded stretched exponential moments. The same framework yields optimal RVE error bounds. The argument works for scalar equations, two-dimensional systems, and systems with Uhlenbeck structure; without (R) only half the rate is obtained. The proof identifies the correctors of the nonlinear problem with those of the linearized problem and exploits the spectral gap inequality to control their stochastic fluctuations.

Load-bearing premise

The load-bearing premise is the spectral gap inequality (P2): every random variable built from the coefficient field must have variance controlled by the ε-scaled integral of its local sensitivity, which encodes that correlations decay fast enough on scales larger than ε.

Editorial extensions

If this is right

  • Numerical simulations of nonlinear random materials can be justified at the same resolution as linear ones: the RVE approximation of the effective material law converges at the central-limit rate (L/ε)^{-d/2} in d≤4.
  • The homogenization error in d≥3 is linear in the microstructural scale, so doubling the scale resolution halves the error—a sharp quantitative target for nonlinear stochastic homogenization.
  • In d=2 a logarithmic correction is unavoidable, matching the critical dimensional behavior of linear elliptic PDEs.
  • For systems with Uhlenbeck structure, or for scalar equations, the same optimal rates hold without assuming extra structure beyond the small-scale regularity condition.
  • The two-scale expansion with correctors also yields an H^1 error estimate for ∇u_ε − ∇û_ε of the same optimal order, making the microscopic gradient approximation quantitative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the spectral gap assumption is relaxed to slower-than-integrable correlations, the paper's strategy predicts that the ε and ε|log ε|^{1/2} rates fail; the rates likely degrade to those governed by the tail behavior of the corrector, analogous to the linear case.
  • The linearized-corrector identification suggests that derivative bounds for the effective operator A_hom(ξ) with respect to ξ might be obtainable by the same machinery, opening a route toward quantitative homogenization of nonlinear elasticity in the small-deformation regime.
  • A natural testable extension is to scalar monotone equations with p-growth (p≠2): the spectral-gap and two-scale-expansion framework would predict some rate ε^α(p), but the present paper does not cover it, and whether α(p) remains 1 for all p is an open question.
  • The RVE error estimates rely on a periodization of the field; for random fields where a matching periodization is hard to construct, the same fluctuation estimates suggest a coupling error that could be controlled by directly comparing local laws, possibly extending the result beyond periodic RVEs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a quantitative stochastic homogenization theory for uniformly elliptic monotone equations and systems with stationary random coefficients that decorrelate on scale ε. Under a spectral-gap inequality and an additional small-scale C^{1,α} regularity condition, it proves homogenization errors of order ε for d≥3 and of order ε|log ε|^{1/2} for d=2 (with a massive term), with random constants having bounded stretched exponential moments. Without the small-scale regularity condition, half rates are claimed, including on bounded domains. The paper also proves RVE approximation error rates of order (L/ε)^{-d/2} for the effective operator, with refined higher-order systematic error estimates. The proof strategy is based on localized correctors and linearized correctors, spectral-gap fluctuation estimates, a two-scale expansion with piecewise affine macroscopic slope, and flux-corrector residual estimates.

Significance. If correct, these results constitute the first optimal-order homogenization rates for a nonlinear stochastic homogenization problem, and the rate-matching to the linear elliptic subfamily makes the optimality argument transparent. The paper has substantial technical strengths: explicit stretched-exponential stochastic moment bounds, a modular proof structure that separates corrector estimates from the two-scale expansion and RVE analysis, and careful treatment of low dimensions and massive terms. The main caveat is that several load-bearing proof ingredients are deferred to a companion paper [21], which makes the manuscript, as submitted, not fully self-contained.

major comments (3)
  1. [3.1, Lemma 18] Lemma 18 is load-bearing: it provides existence, uniqueness, and continuity of the localized correctors and flux correctors on which every subsequent corrector estimate, including Propositions 19 and 20 and hence Theorems 2, 4, 7, and 14, depends. The proof is not contained in the manuscript; the text states 'For a detailed proof see [21]'. This makes the central theorems conditional on an unpublished companion paper. A journal version should either include this proof in full or state the main results as conditional on [21] with the exact dependence made explicit.
  2. [3.1-3.3, two-scale expansion] The passage from the localized correctors with finite T to the actual correctors is handled by Lemma 33, whose proof is only sketched and points to 'the beginning of the proof of Lemma 40'; the argument there is described as a quantitative proof for φ that 'extends to σ', but it is not written out. Since Corollary 21 and the final error estimates rely on this T→∞ limit, the manuscript should provide a complete convergence argument for both the corrector and the flux corrector, or give a precise published reference.
  3. [5.1, proof of Lemma 24, Part c] The estimate for the vector potential θ_T is obtained by omitting the massive regularization of the equation Δθ_T = ∇φ_T; the authors write 'we omit this additional technicality'. This estimate feeds into Proposition 19 and therefore into the d≥3 rates, so the omitted argument is not purely cosmetic. The proof should be included or the status of the omitted regularization should be made precise.
minor comments (5)
  1. [Theorem 14(a)] The displayed moment bound 'EL[exp(C1/C/C)]≤2' appears garbled; it should presumably read E_L[exp((C/C)^{1/C})]≤2 or an equivalent form. Please correct the notation throughout the theorem.
  2. [Lemma 50] In the displayed estimate of Lemma 50, the sum on the right-hand side is written with ∇φ_T but the lemma concerns an arbitrary function v; it should be ∇v. As written, the statement is internally inconsistent, although the proof makes the intended meaning clear.
  3. [1.1 and 2.1] There are several places where u is used instead of u_ε in expressions such as 'The error u−uhom' in Section 1.1. Please standardize the notation.
  4. [Throughout] The symbol C is used both for deterministic constants and for random constants; in formulas such as E[exp(C^{ν}/C)]≤2 this creates ambiguity. Using a distinct notation such as C(ω) for the random constant would improve readability.
  5. [References] The companion paper [21] is cited as forthcoming. If it is already available, the reference should be updated with preprint or publication data; otherwise the dependence of the main theorems on unpublished work should be addressed as in the major comments.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the optimal-rate theorem is derived from the stated spectral-gap and regularity assumptions; the only flagged item is a load-bearing but non-circular deferral of technical proofs to [21].

full rationale

The derivation is self-contained relative to its assumptions (A1)-(A3), (P1)-(P2), and (R). Corrector estimates (Propositions 19-20) are obtained by feeding sensitivities of corrector functionals into the spectral-gap inequality (Lemma 23), then converting functional estimates into Lp corrector bounds via Lemma 32; the homogenization error (Theorems 2, 4, and 7) follows from the two-scale expansion residuum estimate of Proposition 36, whose proof is given in the text. The claim that the rates are optimal because linear elliptic PDEs form a subclass is a legitimate lower-bound argument: an epsilon-rate for the nonlinear class would imply the known optimal linear rate, so it is not circular. The paper explicitly defers two ingredients to a forthcoming paper: the detailed proof of existence of localized correctors ("For a detailed proof see [21]", after Lemma 18) and the formal derivation of the two-scale expansion residual ("we refer to the forthcoming paper [21]", Section 3.1). These deferrals make the proof conditional on [21], but they are missing-support or conditionality concerns, not reductions of a claimed output to an input by construction. No fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported from the authors' prior work to force the ansatz. The self-references to Gloria-Otto and Gloria-Otto-Neukamm for the spectral-gap strategy are methodological background; the nonlinear estimates themselves are derived in the present text. Therefore the appropriate finding is no significant circularity, with a small deduction only for the explicit reliance on unpublished [21] proofs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

All quantitative content is produced from a short list of explicit structural assumptions; no constant is fitted to data and no ad hoc entity is introduced. The price is that the sharpest rates rest on the spectral gap inequality and the small-scale regularity condition, and part of the scaffolding, Lemma 18 and the formal two-scale computation, is deferred to the authors' forthcoming paper [21].

assumptions (6)
  • domain assumption Stationarity of the law of ωε under spatial translation (P1)
    The statistics of the random coefficient field are assumed translation-invariant; used throughout to identify E[∇φ] = 0 and to define Ahom via expectations (Section 2.1, Definition 1).
  • domain assumption Spectral gap inequality with correlation length ε (P2, Definition 16a)
    The quantitative decorrelation hypothesis; all stochastic moment bounds (Lemma 23) and the corrector estimates (Propositions 19-20) flow from it. The paper notes in Section 2.4 that it fails for strongly correlated fields.
  • domain assumption Uniform monotonicity, Lipschitz bounds, A(ω,0)=0, and differentiability in ω (A1)-(A3)
    Structural assumptions on the operator family; they yield existence and uniqueness of solutions and correctors and bound the sensitivity ∂F/∂ωε in the spectral gap estimates.
  • domain assumption Small-scale regularity condition (R): m=1, or d≤2, or Uhlenbeck structure, with ε-scale Lipschitz ωε and bounded ∂²ξA
    Provides the C^{1,α} regularity on the ε scale needed for the linearized corrector estimates (Lemma 31, Propositions 51-52) and for the optimal rates in Theorems 2 and 4. Without it, only half rates are claimed (Theorem 7).
  • domain assumption Existence of an L-periodic approximation PL of P with matching statistics on BL/4 (Definition 13)
    Theorem 14 requires such a periodic approximation; the paper states this must be proven case-by-case and gives a Gaussian convolution example (Section 2.3).
  • standard math Background PDE tools and cited lemmas: Caccioppoli, hole-filling, weighted Meyers, Calderón-Zygmund, Schauder, Gehring; multiscale decomposition Lemma 32 cited from [13]; spectral gap moment bound Lemma 23 cited from [19]; localized corrector existence Lemma 18 sketched with proof deferred to [21]
    The paper proves Lemmas 43-45 in appendices, but relies on prior literature for Lemma 23 and Lemma 32, and defers the detailed proof of Lemma 18 to the authors' forthcoming paper [21].

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Pith. "Pith review of Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems." pith.science (2026). https://pith.science/paper/S7W3YAEZ

@misc{pith2026190802273,
  author       = {Pith},
  title        = {Pith review of: Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7W3YAEZ}},
  note         = {Machine review of arXiv:1908.02273}
}
abstract

We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $\mathbb{R}^d$ with stationary law (i.e. spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $\varepsilon>0$, we establish homogenization error estimates of the order $\varepsilon$ in case $d\geq 3$, respectively of the order $\varepsilon |\log \varepsilon|^{1/2}$ in case $d=2$. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $\varepsilon^\delta$. We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $(L/\varepsilon)^{-d/2}$ for a representative volume of size $L$. Our results also hold in the case of systems for which a (small-scale) $C^{1,\alpha}$ regularity theory is available.

Figures

Figures reproduced from arXiv: 1908.02273 by the authors.

Figure 1
Figure 1. Typical realizations of random fields obtained by ap￾plying a nonlinear map pointwise to a stationary Gaussian random field with only short-range correlations (left), respectively to a sta￾tionary Gaussian random field with barely integrable correlations (right). To state the first example of a random monotone operator satisfying our assump￾tions, consider any two deterministic spatially homogeneous monotone operato… view at source ↗
Figure 2
Figure 2. The structure of the proof of the main results. Note that the dashed arrows each involve an application of Lemma 32 and each result in an estimate for the corresponding minimal radii r∗,ξ respectively r∗,ξ,Ξ (see Lemma 26 and Lemma 30). For sim￾plicity, we have omitted the localization parameter T and the as￾sociated technicalities throughout the diagram. correctors φ T ξ and σ T ξ , which for T > 0 and any paramete… view at source ↗

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Works this paper leans on

45 extracted references · 42 canonical work pages

  1. [21]

    Duerinckx, S

    M. Duerinckx, S. Neukamm, M. Ruf, and M. Sch¨ affner. Quantitative stochastic homogeniza- tion in nonlinear elasticity at small loads. in preparation, 2020

    work page 2020

  2. [1]

    Armstrong and P

    S. Armstrong and P. Cardaliaguet. Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions. J. Eur. Math. Soc. (JEMS) , 20(4):797–864, 2018

    work page 2018

  3. [2]

    Homogenization, linearization and large-scale regularity for nonlinear elliptic equations

    S. Armstrong, S. Ferguson, and T. Kuusi. Homogenization, linearization and large-scale reg- ularity for nonlinear elliptic equations. Preprint, 2018. arXiv:1805.00467

    work page Pith review arXiv 2018

  4. [3]

    Armstrong, S

    S. Armstrong, S. Ferguson, and T. Kuusi. tba. in preparation, 2020

    work page 2020

  5. [4]

    Armstrong, T

    S. Armstrong, T. Kuusi, and J.-C. Mourrat. The additive structure of elliptic homogenization. Invent. Math., 208(3):999–1154, 2017

    work page 2017

  6. [5]

    Quantitative stochastic homogenization and large-scale regularity

    S. Armstrong, T. Kuusi, and J.-C. Mourrat. Quantitative stochastic homogenization and large-scale regularity , volume 352 of Grundlehren der mathematischen Wissenschaften . Springer, 2019. arXiv:1705.05300

    work page Pith review arXiv 2019

  7. [6]

    S. N. Armstrong, P. Cardaliaguet, and P. E. Souganidis. Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations. J. Amer. Math. Soc. , 27(2):479–540, 2014

    work page 2014

  8. [7]

    S. N. Armstrong and J.-C. Mourrat. Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal., 219(1):255–348, 2016

    work page 2016

Show all 45 references
  1. [8]

    S. N. Armstrong and C. K. Smart. Quantitative stochastic homogenization of elliptic equa- tions in nondivergence form. Arch. Ration. Mech. Anal., 214(3):867–911, 2014

    2014

  2. [9]

    S. N. Armstrong and C. K. Smart. Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 49(2):423–481, 2016

    2016

  3. [10]

    Avellaneda and F.-H

    M. Avellaneda and F.-H. Lin. Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. , 40(6):803–847, 1987

    1987

  4. [11]

    J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity.Arch. Rational Mech. Anal., 63(4):337–403, 1976/77

    1976

  5. [12]

    Barchiesi and A

    M. Barchiesi and A. Gloria. New counterexamples to the cell formula in nonconvex homoge- nization. Arch. Ration. Mech. Anal., 195(3):991–1024, 2010

    2010

  6. [13]

    Bella, B

    P. Bella, B. Fehrman, J. Fischer, and F. Otto. Stochastic homogenization of linear elliptic equations: Higher-order error estimates in weak norms via second-order correctors. SIAM J. Math. Anal., 49(6):4658–4703, 2017

    2017

  7. [14]

    Briane and G

    M. Briane and G. A. Francfort. Loss of ellipticity through homogenization in linear elasticity. Math. Models Methods Appl. Sci. , 25(5):905–928, 2015

    2015

  8. [15]

    L. A. Caffarelli and P. E. Souganidis. Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math., 180(2):301–360, 2010

    2010

  9. [16]

    Conti, G

    S. Conti, G. Dolzmann, B. Kirchheim, and S. M¨ uller. Sufficient conditions for the validity of the Cauchy-Born rule close to SO( n). J. Eur. Math. Soc. , 8(3):515–530, 2006

    2006

  10. [17]

    Dal Maso and L

    G. Dal Maso and L. Modica. Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. (4), 144:347–389, 1986

    1986

  11. [18]

    Dal Maso and L

    G. Dal Maso and L. Modica. Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math., 368:28–42, 1986

    1986

  12. [19]

    Duerinckx and A

    M. Duerinckx and A. Gloria. Multiscale functional inequalities in probability: Concentration properties. ALEA, Lat. Am. J. Probab. Math. Stat. , 17:133–157, 2020

    2020

  13. [20]

    Duerinckx and A

    M. Duerinckx and A. Gloria. Multiscale functional inequalities in probability: Constructive approach. to appear in Ann. Henri Lebesgue , 2020. arXiv:1711.03152

    2020 arXiv

  14. [22]

    J. Fischer. The choice of representative volumes in the approximation of effective properties of random materials. Arch. Ration. Mech. Anal., 234(2):635–726, 2019

    2019

  15. [23]

    Fischer and C

    J. Fischer and C. Raithel. Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space. SIAM J. Math. Anal. , 49(1):82–114, 2017

    2017

  16. [24]

    G. A. Francfort and A. Gloria. Isotropy prohibits the loss of strong ellipticity through ho- mogenization in linear elasticity. C. R. Math. Acad. Sci. Paris , 354(11):1139–1144, 2016

    2016

  17. [25]

    Friesecke and F

    G. Friesecke and F. Theil. Validity and failure of the Cauchy-Born hypothesis in a two- dimensional mass-spring lattice. J. Nonlinear Sci. , 12(5):445–478, 2002

    2002

  18. [26]

    Geymonat, S

    G. Geymonat, S. M¨ uller, and N. Triantafyllidis. Homogenization of non-linearly elastic ma- terials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rational Mech. Anal., 122(3):231–290, 1993. 100 JULIAN FISCHER AND STEFAN NEUKAMM

    1993

  19. [27]

    Giaquinta and L

    M. Giaquinta and L. Martinazzi. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, volume 11. Edizioni della Normale, Pisa, second edition, 2012

    2012

  20. [28]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order . Springer, 2001

    2001

  21. [29]

    Gloria, S

    A. Gloria, S. Neukamm, and F. Otto. Quantification of ergodicity in stochastic homogeniza- tion: optimal bounds via spectral gap on Glauber dynamics. Invent. Math., 199(2):455–515, 2015

    2015

  22. [30]

    Gloria, S

    A. Gloria, S. Neukamm, and F. Otto. Quantitative estimates in stochastic homogenization for correlated coefficient fields. Analysis & PDE , 2020. to appear

    2020

  23. [31]

    Gloria, S

    A. Gloria, S. Neukamm, and F. Otto. A regularity theory for random elliptic operators. Milan J. Math., 88:99–170, 2020

    2020

  24. [32]

    Gloria and F

    A. Gloria and F. Otto. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab., 39(3):779–856, 2011

    2011

  25. [33]

    Gloria and F

    A. Gloria and F. Otto. An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab., 22(1):1–28, 2012

    2012

  26. [34]

    Gloria and F

    A. Gloria and F. Otto. The corrector in stochastic homogenization: Near-optimal rates with optimal stochastic integrability. Preprint, 2015. arXiv:1510.08290

    2015 arXiv

  27. [35]

    Gloria and M

    A. Gloria and M. Ruf. Loss of strong ellipticity through homogenization in 2D linear elasticity: a phase diagram. Arch. Ration. Mech. Anal., 231(2):845–886, 2019

    2019

  28. [36]

    Guti´ errez

    S. Guti´ errez. Laminations in linearized elasticity: the isotropic non-very strongly elliptic case. J. Elasticity, 53(3):215–256, 1998/99

    1998

  29. [37]

    Mitake, H

    H. Mitake, H. V. Tran, and Y. Yu. Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting. Arch. Ration. Mech. Anal. , 233(2):901–934, 2019

    2019

  30. [38]

    Naddaf and T

    A. Naddaf and T. Spencer. Estimates on the variance of some homogenization problems. Unpublished preprint, 1998

    1998

  31. [39]

    Neff and D

    P. Neff and D. Knees. Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity. SIAM J. Math. Anal. , 40(1):21–43, 2008

    2008

  32. [40]

    Neukamm and M

    S. Neukamm and M. Sch¨ affner. Quantitative homogenization in nonlinear elasticity for small loads. Arch. Ration. Mech. Anal., 230(1):343–396, 2018

    2018

  33. [41]

    Neukamm and M

    S. Neukamm and M. Sch¨ affner. Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains.Calc. Var. Partial Differential Equations, 58(2):Art. 46, 51, 2019

    2019

  34. [42]

    Uhlenbeck

    K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. Acta Math., 138(3-4):219– 240, 1977

    1977

  35. [43]

    L. Wang, Q. Xu, and P. Zhao. Quantitative estimates on periodic homogenization of nonlinear elliptic operators. Preprint, 2018. arXiv:1807.10865

    2018 arXiv

  36. [44]

    L. Wang, Q. Xu, and P. Zhao. Convergence rates on periodic homogenization of p-Laplace type equations. Nonlinear Anal. Real World Appl. , 49:418–459, 2020

    2020

  37. [45]

    K. Zhang. Energy minimizers in nonlinear elastostatics and the implicit function theorem. Arch. Rational Mech. Anal., 114(2):95–117, 1991. (J. Fischer) IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria (S. Neukamm) TU Dresden, F aculty of Mathematics, 01062 Dresden

    1991

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