Homogenization and boundary layers

Gérard-Varet, David; Masmoudi, Nader

doi:10.1007/s11511-012-0083-5

Cited by 76 publications

(117 citation statements)

References 26 publications

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“…In the periodic case, the analysis for general domains requires either a careful geometric analysis, such as in [12], to obtain a uniform modulus of continuity, or an argument as in [17] that ignores discontinuities of the data in small parts of the boundary. We also point out the recent results of Gérard-Varet and Masmoudi [18,19] about systems of divergence-form operators with oscillatory Dirichlet data in periodic media with error estimates. In a similar vein are the results of Kenig, Lin and Shen [24] on the rate of convergence for interior homogenization with Dirichlet or Neumann boundary conditions in Lipschitz domains.…”

Section: Introductionmentioning

confidence: 79%

Quantitative homogenization of elliptic partial differential equations with random oscillatory boundary data

Feldman

Kim

Souganidis

2015

Journal de Mathématiques Pures et Appliquées

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We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media leading to concentration of measure, we prove an almost sure and local uniform homogenization result with a rate of convergence in probability.© 2014 Elsevier Masson SAS. All rights reserved.r é s u m éOn étudie le comportement homogénéisant d'équations aux dérivées partielles elliptiques nonlinéaires, avec conditions au bord de Dirichlet ou de Neumann aléatoires, oscillantes à petite échelle. Sous certaines contraintes sur l'opérateur, telles que les données et les milieux aléatoires conduisent à une concentration de la mesure, on démontre un résultat presque sûr d'homogénéisation locale uniforme, avec un taux de convergence en probabilité.

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Section: Introductionmentioning

confidence: 79%

Quantitative homogenization of elliptic partial differential equations with random oscillatory boundary data

Feldman

Kim

Souganidis

2015

Journal de Mathématiques Pures et Appliquées

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“…To the best knowledge of the author not even continuous dependency on α has yet been established. However, in the case of boundary layers in continuum homogenization recent results in this direction have been obtained in [14].…”

Section: Description Of the Model And Main Resultsmentioning

confidence: 99%

Surface energies in a two-dimensional mass-spring model for crystals

Theil¹

2011

ESAIM: M2AN

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Abstract.We study an atomistic pair potential-energy E (n) (y) that describes the elastic behavior of two-dimensional crystals with n atoms where y ∈ R 2×n characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E (n) admits an asymptotic expansion involving fractional powers of n:The bulk energy density E bulk is given by an explicit expression involving the interaction potentials. The surface energy E surface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal.Mathematics Subject Classification. 74Q05.

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“…Boundary layers need to be taken into account. Considering 11) proves to be more relevant than (1.2). Moreover, ϑ * bl , appearing in (1.10), comes from the homogenization of an elliptic boundary layer system like (1.11).…”

Section: Homogenization Of Boundary Layer Systemsmentioning

confidence: 95%

“…(1) convex polygonal domains Ω, first with edges of rational slopes by Moskow and Vogelius in [17], Allaire and Amar in [3] (scalar case), then with edges of slopes satisfying a generic small divisors assumption by Gérard-Varet and Masmoudi in [10]; (2) smooth domains with uniformly convex boundary by Gérard-Varet and Masmoudi in the recent paper [11].…”

Section: Homogenization Of Boundary Layer Systemsmentioning

confidence: 99%

First-order expansion for the Dirichlet eigenvalues of an elliptic system with oscillating coefficients

Prange

2013

Asymptotic Analysis

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This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω ⊂ R 2 , for a vectorial elliptic operator −∇ · A ε (·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λ ε,k when ε → 0, the mode k being fixed. A first-order asymptotic expansion is proved for λ ε,k in the case when Ω is either a smooth uniformly convex domain, or a convex polygonal domain with sides of slopes satisfying a small divisors assumption. Our results extend those of Moskow and Vogelius in restricted to scalar operators and convex polygonal domains with sides of rational slopes. We take advantage of the recent progress due to Gérard-Varet and Masmoudi [J.

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Homogenization and boundary layers

Cited by 76 publications

References 26 publications

Quantitative homogenization of elliptic partial differential equations with random oscillatory boundary data

Quantitative homogenization of elliptic partial differential equations with random oscillatory boundary data

Surface energies in a two-dimensional mass-spring model for crystals

First-order expansion for the Dirichlet eigenvalues of an elliptic system with oscillating coefficients

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