Uniform boundary regularity in almost-periodic homogenization

Zhuge, Jinping

doi:10.1016/j.jde.2016.09.031

Cited by 8 publications

(4 citation statements)

References 20 publications

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“…Another recent breakthrough was made by S. Armstrong and Z. Shen in [2] for the almost-periodic setting, and they developed a new method which based on convergence rates rather than the compactness methods. We refer the reader to [1,19,26] and its reference therein for more details on non-periodic cases. Meanwhile, T. Suslina [20,21] obtained the sharp O(ε) convergence rates in L 2 (Ω) for elliptic homogenization problems in C 1,1 domains, while C. Kenig, F. Lin, and Z. Shen [15] figured out the almost sharp one O(ε ln(1/ε)) concerned with Lipschitz domains, and their results have been improved by the second author in [23], recently.…”

Section: Instruction and Main Resultsmentioning

confidence: 99%

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

Gu¹,

Xu²

2016

Preprint

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The paper concerns the sharp boundary regularity estimates in homogenization of Dirichlet problem for Stokes systems. We obtain the Lipschitz estimates for velocity term and L ∞ estimate for pressure term, under some reasonable smoothness assumption on rapidly oscillating periodic coefficients. The approach is based on convergence rates, originally investigated by S. Armstrong and Z. Shen in [2,17], however the argument developed here does not rely on the Rellich estimates. In this sense, we find a new way to obtain the sharp uniform boundary estimates without imposing the symmetry assumption on coefficients. Additionally, we emphasize that L ∞ estimate for the pressure term does require the O(ε 1/2 ) convergence rate, locally at least, compared to O(ε λ ) for the velocity term, where λ ∈ (0, 1/2).

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Section: Instruction and Main Resultsmentioning

confidence: 99%

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

Gu¹,

Xu²

2016

Preprint

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“…The approach was further developed in [3,35], where the large-scale interior or boundary Lipschitz estimates for second-order elliptic operators with periodic and almost periodic coefficients were studied systematically. We also refer readers to [2,16,17,46] for more related results.…”

Section: Introductionmentioning

confidence: 99%

Uniform boundary estimates in homogenization of higher-order elliptic systems

Niu

2018

Annali di Matematica

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This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators L ε , with rapidly oscillating periodic coefficients. We derive uniform boundary C m−1,λ (0 <λ<1) and W m, p estimates in C 1 domains, as well as uniform boundary C m−1,1 estimate in C 1,θ (0 <θ <1) domains without the symmetry assumption on the operator. The proof, motivated by the works "Armstrong and Smart in

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“…This approach was further developed in [5,34] for second order elliptic systems with periodic and almost periodic coefficients. Using this method, the large scale interior or boundary Lipschitz estimates for second order elliptic operators were studied [6,5,34], see also [18,17,4,43,3] for more related results.…”

Section: Introductionmentioning

confidence: 99%

Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems

Niu

2017

Preprint

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This paper focuses on the uniform boundary estimates in homogenization of a family of higher order elliptic operators L ε , with rapidly oscillating periodic coefficients. We derive uniform boundary C m−1,λ (0 < λ < 1), W m,p estimates in C 1 domains, as well as uniform boundary C m−1,1 estimate in C 1,θ (0 < θ < 1) domains without the symmetry assumption on the operator. The proof, motivated by the profound work "S.N. Armstrong and C. K. Smart, Ann. Sci. Éc. Norm. Supér. ( 2016), Z. Shen, Anal. PDE (2017)", is based on a suboptimal convergence rate in H m−1 (Ω). Compared to "C.E. Kenig, F. Lin and Z. Shen, Arch. Ration. Mech. Anal. (2012), Z. Shen, Anal. PDE ( 2017)", the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of u 0 (the solution to the homogenized problem), and thus might be of some independent interests even for second order elliptic systems.

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Uniform boundary regularity in almost-periodic homogenization

Cited by 8 publications

References 20 publications

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

Uniform boundary estimates in homogenization of higher-order elliptic systems

Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems

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