Averaging of Random Operators

Козлов, С. М.

doi:10.1070/sm1980v037n02abeh001948

Cited by 217 publications

(254 citation statements)

References 11 publications

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“…The qualitative theory of stochastic homogenization dates back to the seminal contributions of Papanicolaou and Varadhan [14], and Kozlov [9]. Let D a bounded domain of R d , f ∈ H −1 (D), and A be a stationary ergodic random field.…”

Section: Introductionmentioning

confidence: 99%

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

Gloria

Otto

2015

ESAIM: Proc.

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Abstract. We establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for d > 2. This work is based on [5], which is a complete continuum version of [6,7] (with in addition optimal results for d = 2). The main difference with respect to the first part of [5] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.Résumé. Nous démontrons des résultats quantitatifs sur l'approximation par périodisation du correcteur en homogénéisation stochastique deséquations aux dérivées partielles elliptiques linéaires sous forme divergence, lorsque les coefficients de diffusion satisfont une hypothèse de trou spectral en probabilité et en dimension d > 2. Ce travail s'inspire de [5], qui est une version complète dans le cas continu de [6,7] (qui traiteégalement le cas d = 2 de manière optimale). La différence majeure avec [5] est qu'on utilise directement la théorie de De Giorgi-Nash-Moserà la place des fonctions de Green.

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Section: Introductionmentioning

confidence: 99%

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

Gloria

Otto

2015

ESAIM: Proc.

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“…The homogenization of elliptic equations in random media originated in the work of Papanicolaou and Varadhan [10,11] and Kozlov [8,9] about three decades ago. Linear equations are somewhat simpler to analyze since they possess a dual structure.…”

Section: Moreover There Exists a Modulusmentioning

confidence: 99%

Stochastic homogenization of fully nonlinear uniformly elliptic equations revisited

Armstrong

Smart

2013

Calc. Var.

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Abstract. We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In the latter case, we overcome difficulties caused by a lack of estimates for the first derivatives of approximate correctors by modifying the perturbed test function argument to take advantage of the spreading of the contact set.

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“…e.g. [15]), and was adapted by Kozlov for stochastic homogenization problems [12,14]. For other stochastic approaches we refer to [5,8,17].…”

Section: Theorem 11 (Homogenization)mentioning

confidence: 99%

Homogenization of the Prager model in one-dimensional plasticity

Schweizer

2009

Continuum Mech. Thermodyn.

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We propose a new method for the homogenization of hysteresis models of plasticity. For the one-dimensional wave equation with an elasto-plastic stress-strain relation we derive averaged equations and perform the homogenization limit for stochastic material parameters. This generalizes results of the seminal paper by Franců and Krejčí. Our approach rests on energy methods for partial differential equations and provides short proofs without recurrence to hysteresis operator theory. It has the potential to be extended to the higher dimensional case.

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Averaging of Random Operators

Cited by 217 publications

References 11 publications

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

Stochastic homogenization of fully nonlinear uniformly elliptic equations revisited

Homogenization of the Prager model in one-dimensional plasticity

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