Homogenization of the boundary value for the Neumann problem

Zhao, Jie

doi:10.1063/1.4909526

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…In 2014, Kenig, Lin, and Shen [14] also obtained W k,p convergence rates of Dirichlet or Neumann problems for the second-order equations with rapidly oscillating periodic coefficients by using the asymptotic estimates of the Green or Neumann functions. In 2015, the second author [24] proved the pointwise as well as W 1,p convergence estimates for the fixed operators and oscillating Neumann boundary data by utilizing oscillation integral estimates in Fourier analysis. In 2016, Shen [17] proved the L q convergence rates with Dirichlet or Neumann problems with no smoothness assumption on the coefficients.…”

Section: )mentioning

confidence: 99%

Convergence rates of nonlinear Stokes problems in homogenization

Wang

Zhao

2019

Bound Value Probl

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In this paper, we study the convergence rates of solutions in homogenization of nonlinear Stokes Dirichlet problems. The main difficulty of this work is twofold. On the one hand, the nonlinear Stokes problems do not fit the standard framework of second-order elliptic equations in divergence form. On the other hand, nonlinear problems may cause new difficulties in the estimation of the quantity as well as first-order approximate term. As a consequence, we establish the sharp rates of convergence in H 1 and L 2. This work may be regarded as an extension of the approach for the linear Stokes problems to the nonlinear case.

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Section: )mentioning

confidence: 99%

Convergence rates of nonlinear Stokes problems in homogenization

Wang

Zhao

2019

Bound Value Probl

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“…In 2014, Kenig, Lin and Shen [8] established W k,p convergence rates, via the asymptotic behavior of the Green or Neumann functions. In 2015, the first author [24] obtained the pointwise as well as W 1,p convergence rates for fixed operators and oscillating Neumann boundary data. In 2015, Gu [5] also proved convergence rates in L 2 and H 1 for linear Stokes systems.…”

Section: Introductionmentioning

confidence: 99%

Convergence rates in homogenization of p-Laplace equations

Zhao

Wang

2019

Bound Value Probl

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This paper is concerned with homogenization of p-Laplace equations with rapidly oscillating periodic coefficients. The main difficulty of this work is due to the nonlinear structure in this field concerning p-Laplace equations itself. Utilizing the layer and co-layer type estimates as well as homogenization techniques, we establish the desired error estimates. As a consequence, we obtain the rates of convergence for solutions in W 1,p 0 as well as L p . Meanwhile, our convergence rate results do not involve the higher derivatives any more. This may be viewed as rather surprising. The novelty of this work is that it seems to find a new analysis method in quantitative homogenization. MSC: Primary 35B27; secondary 35J15

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“…In 2014, they [5] have also studied the asymptotic behavior of the Green and Neumann functions obtaining some error estimates of solutions. In 2015, the first author [6] obtained the pointwise as well as 1, convergence results, which is based on Fourier analysis. In 2016, Shen [7] proved the 1 convergence rates with Dirichlet or Neumann conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1. Suppose that ∈ 1 (Ω) and 0 ∈ 2 (Ω) are the weak solutions of the mixed boundary value problems (1) and (6), respectively. Then, under the assumptions (2)- (5), there exists a constant C such that…”

Section: Introductionmentioning

confidence: 99%