Averaging Differential Operators With Almost Periodic, Rapidly Oscillating Coefficients

Козлов, С. М.

doi:10.1070/sm1979v035n04abeh001561

Cited by 105 publications

(92 citation statements)

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“…In the almost periodic setting, our results are new. In fact, till now the only results in that setting pertain to Braides [6] and to Kozlov [15] who considered, for the first author, the almost periodic homogenization of functionals of the same type as Allaire (see here above) with f being almost periodic in its first occurrence, and for the second one, the almost periodic homogenization of functionals of the same type, but with f being a quadratic form in its second occurrence. Apart from these two problems (the periodic and the almost periodic ones) all the other problems involved in this study lead to new results.…”

Section: Resultsmentioning

confidence: 99%

“…In [1], under a periodic hypothesis, and a differentiability hypothesis on f with respect to λ, Allaire recovers some explicit results via the technique of two-scale convergence. Concerning the results beyond the periodic setting, Kozlov [15] was the first to prove an homogenization result for functionals of the form (1.7), with f being a quadratic form in its second occurrence and almost periodic in the first one. Later on, Braides [6], using the Γ-convergence techniques, studied functionals (1.7) under the almost periodicity assumption with respect to the fast variable y and the quasiconvexity hypothesis on f with respect to the second variable.…”

Section: Introductionmentioning

confidence: 99%

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Deterministic homogenization of integral functionals with convex integrands

Nguetseng

Nnang

Woukeng

2010

Nonlinear Differ. Equ. Appl.

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Abstract. In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence. Mathematics Subject Classification (2000). 35B40, 45G10, 46J10.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deterministic homogenization of integral functionals with convex integrands

Nguetseng

Nnang

Woukeng

2010

Nonlinear Differ. Equ. Appl.

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“…We refer the reader to [12] for details (also see earlier work in [15,16,17]). The proof of the following theorem may be found in [12].…”

Section: The Homogenized Operator and Qualitative Homogenizationmentioning

confidence: 99%

Lipschitz Estimates in Almost‐Periodic Homogenization

Armstrong

Shen

2015

Comm Pure Appl Math

120

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We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C 1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin [2,5] for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen [14] for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat non-symmetric coefficients. We also obtain uniform W 1,p estimates.

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“…The qualitative theory of stochastic homogenization for linear elliptic equations in divergence form was completed in the early 1980s by Papanicolaou and Varadhan [36], Kozlov [27], Yurinskii [40] and, later, using variational methods, by Dal Maso and Modica [11,12]. Each of these results state roughly that, P-almost surely, solutions of −∇ ⋅ (a(x)∇u) = 0 in U converge in L 2 (U ) (with L 2 suitably normalized relatively to the size of U ) as the domain U becomes large to those of a deterministic, constant-coefficient equation −∇ ⋅ (a∇u) = 0.…”

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confidence: 99%

The additive structure of elliptic homogenization

2016

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Abstract. One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in [2]: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

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Averaging Differential Operators With Almost Periodic, Rapidly Oscillating Coefficients

Cited by 105 publications

References 1 publication

Deterministic homogenization of integral functionals with convex integrands

Deterministic homogenization of integral functionals with convex integrands

Lipschitz Estimates in Almost‐Periodic Homogenization

The additive structure of elliptic homogenization

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