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November  2013, 12(6): 2787-2795. doi: 10.3934/cpaa.2013.12.2787

Convergence rates for elliptic reiterated homogenization problems

Jie Zhao 1,

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received  March 2012 Revised  September 2012 Published  May 2013

In this paper, we study the convergence rates for the reiterated homogenization for equations of the form $-div(A(\frac{x}{\varepsilon},\frac{x}{\varepsilon^{2}})\nabla u_{\varepsilon})=f(x)$. As a consequence, we obtain the convergence rates in $L^{p}$ for solutions with Dirichlet boundary condition by a method based on the representation of elliptic equation solution by Green function. Meanwhile, the growth rate of Green function is found.
Keywords: Green function., convergence rates, Homogenization.
Mathematics Subject Classification: Primary: 35J15; Secondary: 35J2.
Citation: Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
References:
[1]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in North-Holland, 1978. doi: 10.1115/1.3424588.  Google Scholar

[2]

M. Avellaneda and F. H. Lin, Homogenization of elliptic problems with $L^p$ boundary date, Appl. Math. Optimization, 15 (1987), 93-107. doi: 10.1007/BF01442648.  Google Scholar

[3]

M. Avellaneda and F. H. Lin, Compactness methods in the thoery of homogenization, Comm. Pure. Appl. Math., 40 (1987), 803-847. doi: 10.1002/cpa.3160400607.  Google Scholar

[4]

M. Avellaneda and F. H. Lin, Compactness methods in the thoery of homogenization $\Pi$: Equations in non-divergence form, Comm. Pure. Appl. Math., 42 (1989), 139-172. doi: 10.1002/cpa.3160420203.  Google Scholar

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M. Avellaneda and F. H. Lin, $L^p$ bounds on singular integral in homogenization, Comm. Pure. Appl. Math., 44 (1991), 897-910. doi: 10.1002/cpa.3160440805.  Google Scholar

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C. E. Kenig, F. H. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems,, preprint, ().   Google Scholar

[7]

C. E. Kenig, F. H. Lin and Z. Shen, Periodic homogenization of Green function and Neumann functions,, preprint, ().   Google Scholar

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D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equation of Second Order," Springer-Verlag, Heidel berg, New York, (1998). doi: 10.1007/978-3-642-61798-0.  Google Scholar

show all references

References:
[1]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in North-Holland, 1978. doi: 10.1115/1.3424588.  Google Scholar

[2]

M. Avellaneda and F. H. Lin, Homogenization of elliptic problems with $L^p$ boundary date, Appl. Math. Optimization, 15 (1987), 93-107. doi: 10.1007/BF01442648.  Google Scholar

[3]

M. Avellaneda and F. H. Lin, Compactness methods in the thoery of homogenization, Comm. Pure. Appl. Math., 40 (1987), 803-847. doi: 10.1002/cpa.3160400607.  Google Scholar

[4]

M. Avellaneda and F. H. Lin, Compactness methods in the thoery of homogenization $\Pi$: Equations in non-divergence form, Comm. Pure. Appl. Math., 42 (1989), 139-172. doi: 10.1002/cpa.3160420203.  Google Scholar

[5]

M. Avellaneda and F. H. Lin, $L^p$ bounds on singular integral in homogenization, Comm. Pure. Appl. Math., 44 (1991), 897-910. doi: 10.1002/cpa.3160440805.  Google Scholar

[6]

C. E. Kenig, F. H. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems,, preprint, ().   Google Scholar

[7]

C. E. Kenig, F. H. Lin and Z. Shen, Periodic homogenization of Green function and Neumann functions,, preprint, ().   Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equation of Second Order," Springer-Verlag, Heidel berg, New York, (1998). doi: 10.1007/978-3-642-61798-0.  Google Scholar

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