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

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.
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.

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.

[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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