Convergence rates for elliptic reiterated homogenization problems

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

Convergence rates for elliptic reiterated homogenization problems

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

Received March 2012 Revised September 2012 Published May 2013

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	C. E. Kenig, F. H. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems,, preprint, ().
[7]	C. E. Kenig, F. H. Lin and Z. Shen, Periodic homogenization of Green function and Neumann functions,, preprint, ().
[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.

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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.
[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.
[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.
[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.
[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.
[6]	C. E. Kenig, F. H. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems,, preprint, ().
[7]	C. E. Kenig, F. H. Lin and Z. Shen, Periodic homogenization of Green function and Neumann functions,, preprint, ().
[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.

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