Abstract
In this paper, we study reiterated homogenization problems for equations −div(A(x/ε, x/ε2)∇uε) = f (x). We introduce auxiliary functions and obtain the representation formula satisfied by uε and homogenized solution u0. Then we utilize this formula in combination with the asymptotic estimates of Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in Lp for solutions as well as gradient error estimates for Neumann problems.
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References
G. Allaire, Shape optimization by the homogenization method, Springer-Verlag, Heidel berg, New York, 2002.
G. Allaire and M. Briane, Multiscale divergence and reiterated homogenization, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297–342.
M. Avellaneda and F. H. Lin, Homogenization of elliptic problem with Lp boundary data, Appl. Math. Optimization, 15 (1987), 93–107.
M. Avellaneda and F. H. Lin, Compactness methods in the theory of homogenization, Comm. Pure. Appl. Math., 40 (1987), 803–847.
A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Studies in North-Holland, 1978.
A. Bradies and D. Lukkassen, Reiterated homogenization of integral functionals, Math. Models Methods Appl., 10 (2000), 1–25.
A. Braicles and A. Defranceschi, Homogenization of multiple integrals, Clarenoon, Oxford, 1998.
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Heidel berg, New York, 1998.
D. Gioranescu, J. Paulin, and J. Saint, Homogenization of reticulated structures, Springer-Verlag, Heidel berg, New York, 1999.
D. Gioranescu and P. Donato, An introduction to homogenization, Oxford University, 1999.
G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269–286.
G. Griso, Interior error estimate for periodic homogenization, Anal. Appl., 4 (2006), 61–79.
U. Hornung, Homogenization and porous media, Springer-Verlag, Heidel berg, New York, 1997.
V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenziation of differnetial operators and integral functions, Springer-Verlag, Berlin, (1994).
C. E. Kenig, F. H. Lin, and Z. W. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc., 26 (2013), 901–937.
C. E. Kenig, F. H. Lin, and Z. W. Shen, Convergence rates in L2 for elliptic homogenization problems, Arch. Rational Mech. Anal., 203 (2012), 1009–1036.
C. E. Kenig, F. H. Lin, and Z. W. Shen, Periodic homogenization of Green and Neumann functions, Comm. Pure. Appl. Math., 67 (2014) 1–44.
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1–68.
J. L. Lions, D. Lukkassen, L. E. Persson, and P. Wall, Reiterated homogenization ofmonotone operators, C. R. Acad. Sci. Paris, 330 (2000), 675–680.
I. V. Andrianov, L. I. Manevitch, and V. G. Oshmyan, Mechanics of periodically heterogeneous structures, Springer-Verlag, Heidel berg, New York, 2002.
N. Meunier and J. V. Schaftingen, Reiterated homogenization for elliptic operators, C. R. Acad. Sci. Paris, 340 (2005), 209–214.
N. Meunier and J. V. Schaftingen, Periodic reiterated homogenization for elliptic functions, J. Math. Pures Appl., 84 (2005), 1716–1743.
O. A. Oleinik, A. S. Shameaev, and G. A. Yosifian, Mathematical problems in elestricty and homogenization, North-Holland, Amesterdam, (1992).
S. E. Pastukhova, The Neumann problem for elliptic equations with multiscale coefficients:operator estimates for homogenization, SB. Math., 207 (2016), 418–443.
E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University, New Jersey 1970.
Acknowledgement
The author would like to thank the reviewers for their valuable comments and helpful suggestions to improve the quality of this paper. This work has been supported by Natural Science Foundation of China (No. 11626239), China Scholarship Council (No. 201708410483), as well as Foundation of Education Department of Henan Province (No. 18A110037). The part of this work was done while the author was visiting School of Mathematics and Applied Statistics, University of Wollongong, Australia.
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Wang, J., Zhao, J. Convergence Rates of Solutions for Elliptic Reiterated Homogenization Problems. Indian J Pure Appl Math 51, 839–856 (2020). https://doi.org/10.1007/s13226-020-0435-3
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DOI: https://doi.org/10.1007/s13226-020-0435-3