April  2019, 39(4): 2295-2323. doi: 10.3934/dcds.2019097

Periodic homogenization of elliptic systems with stratified structure

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

2. 

School of Mathematical Science, Anhui University, Hefei 230601, China

* Corresponding author: Weisheng Niu

Received  September 2018 Revised  October 2018 Published  January 2019

Fund Project: The first author is supported by the NSF of China (11731005, 11801227, 11801228), and the second anthor is supported by the NSF of China (11701002) and NSF of Anhui Province (1708085MA02)

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp $ O(\varepsilon) $-convergence rate in $ L^{p_0}(\Omega) $ with $ p_0 = \frac{2d}{d-1} $ is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an $ O(\varepsilon^\sigma) $-convergence rate is also derived for some $ \sigma<1 $ by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior $ W^{1, p} $ and Hölder estimates are also obtained by the real variable method.

Citation: Yao Xu, Weisheng Niu. Periodic homogenization of elliptic systems with stratified structure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2295-2323. doi: 10.3934/dcds.2019097
References:
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G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar

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S. N. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923. doi: 10.1002/cpa.21616. Google Scholar

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S. N. Armstrong and C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 423-481. doi: 10.24033/asens.2287. Google Scholar

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

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

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M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization. Ⅱ. Equations in nondivergence form, Comm. Pure Appl. Math., 42 (1989), 139-172. doi: 10.1002/cpa.3160420203. Google Scholar

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R. BunoiuG. Cardone and T. Suslina, Spectral approach to homogenization of an elliptic operator periodic in some directions, Math. Methods Appl. Sci., 34 (2011), 1075-1096. doi: 10.1002/mma.1424. Google Scholar

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L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G. Google Scholar

[13]

R. DongD. Li and L. Wang, Regularity of elliptic systems in divergence form with directional homogenization, Discrete Contin. Dyn. Syst., 38 (2018), 75-90. doi: 10.3934/dcds.2018004. Google Scholar

[14]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems, in Multiscale Methods in Science and Engineering, vol. 44 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2005, 89-110. doi: 10.1007/3-540-26444-2_4. Google Scholar

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W. EP. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

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J. Geng, $W^{1,p}$ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math., 229 (2012), 2427-2448. doi: 10.1016/j.aim.2012.01.004. Google Scholar

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J. GengZ. Shen and L. Song, Uniform $W^{1,p}$ estimates for systems of linear elasticity in a periodic medium, J. Funct. Anal., 262 (2012), 1742-1758. doi: 10.1016/j.jfa.2011.11.023. Google Scholar

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M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1983. Google Scholar

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B. GustafssonJ. Mossino and C. Picard, $H$-convergence for stratified structures with high conductivity, Adv. Math. Sci. Appl., 4 (1994), 265-284. Google Scholar

[20]

B. Heron and J. Mossino, $H$-convergence and regular limits for stratified media with low and high conductivities, Appl. Anal., 57 (1995), 271-308. doi: 10.1080/00036819508840352. Google Scholar

[21]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. Google Scholar

[22]

C. E. KenigF. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal., 203 (2012), 1009-1036. doi: 10.1007/s00205-011-0469-0. Google Scholar

[23]

C. E. KenigF. Lin and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc., 26 (2013), 901-937. doi: 10.1090/S0894-0347-2013-00769-9. Google Scholar

[24]

C. E. Kenig and Z. Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math., 64 (2011), 1-44. doi: 10.1002/cpa.20343. Google Scholar

[25]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189-206. Google Scholar

[26]

W. NiuZ. Shen and Y. Xu, Convergence rates and interior estimates in homogenization of higher order elliptic systems, J. Funct. Anal., 274 (2018), 2356-2398. doi: 10.1016/j.jfa.2018.01.012. Google Scholar

[27]

Z. Shen, Bounds of Riesz transforms on $L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier, 55 (2005), 173-197. doi: 10.5802/aif.2094. Google Scholar

[28]

Z. Shen, The $L^p$ boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254. doi: 10.1016/j.aim.2007.05.017. Google Scholar

[29]

Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653. Google Scholar

[30]

Z. Shen, Periodic Homogenization of Elliptic Systems, Advances in Partial Differential Equations, No. 269, Birkhuser Basel, 2018. doi: 10.1007/978-3-319-91214-1. Google Scholar

[31]

Z. Shen and J. Zhuge, Convergence rates in periodic homogenization of systems of elasticity, Proc. Amer. Math. Soc., 145 (2017), 1187-1202. doi: 10.1090/proc/13289. Google Scholar

[32]

S. Shkoller, An approximate homogenization scheme for nonperiodic materials, Comput. Math. Appl., 33 (1997), 15-34. doi: 10.1016/S0898-1221(97)00003-5. Google Scholar

[33]

T. A. Suslina, On the averaging of a periodic elliptic operator in a strip, Algebra i Analiz, 16 (2004), 269-292. doi: 10.1090/S1061-0022-04-00849-0. Google Scholar

[34]

T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: $L^2$-operator error estimates, Mathematika, 59 (2013), 463-476. doi: 10.1112/S0025579312001131. Google Scholar

[35]

T. A. Suslina, Homogenization of the Neumann problem for elliptic systems with periodic coefficients, SIAM J. Math. Anal., 45 (2013), 3453-3493. doi: 10.1137/120901921. Google Scholar

[36]

D. TsalisT. BaxevanisG. Chatzigeorgiou and N. Charalambakis, Homogenization of elastoplastic composites with generalized periodicity in the microstructure, International Journal of Plasticity, 51 (2013), 161-187. doi: 10.1016/j.ijplas.2013.05.006. Google Scholar

[37]

D. TsalisG. Chatzigeorgiou and N. Charalambakis, Homogenization of structures with generalized periodicity, Composites Part B: Engineering, 43 (2012), 2495-2512. doi: 10.1016/j.compositesb.2012.01.054. Google Scholar

[38]

F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math., 67 (1945), 83-93. doi: 10.2307/2371917. Google Scholar

[39]

Q. Xu, Convergence rates and $W^{1,p}$ estimates in homogenization theory of Stokes systems in Lipschitz domains, J. Differential Equations, 263 (2017), 398-450. doi: 10.1016/j.jde.2017.02.040. Google Scholar

[40]

Y. Xu and W. Niu, Convergence rates in almost-periodic homogenization of higher-order elliptic systems, preprint, arXiv: 1712.01744, 1-44.Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar

[2]

S. N. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923. doi: 10.1002/cpa.21616. Google Scholar

[3]

S. N. Armstrong and C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 423-481. doi: 10.24033/asens.2287. Google Scholar

[4]

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

[5]

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

[6]

M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization. Ⅱ. Equations in nondivergence form, Comm. Pure Appl. Math., 42 (1989), 139-172. doi: 10.1002/cpa.3160420203. Google Scholar

[7]

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

[8]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1978. Google Scholar

[9]

M. Briane, Homogénéisation de Matériaux Fibrés et Multi-couches, PhD Thesis, University Paris 6, Paris, 1990.Google Scholar

[10]

M. Briane, Three models of nonperiodic fibrous materials obtained by homogenization, RAIRO Modél. Math. Anal. Numér., 27 (1993), 759-775. doi: 10.1051/m2an/1993270607591. Google Scholar

[11]

R. BunoiuG. Cardone and T. Suslina, Spectral approach to homogenization of an elliptic operator periodic in some directions, Math. Methods Appl. Sci., 34 (2011), 1075-1096. doi: 10.1002/mma.1424. Google Scholar

[12]

L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G. Google Scholar

[13]

R. DongD. Li and L. Wang, Regularity of elliptic systems in divergence form with directional homogenization, Discrete Contin. Dyn. Syst., 38 (2018), 75-90. doi: 10.3934/dcds.2018004. Google Scholar

[14]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems, in Multiscale Methods in Science and Engineering, vol. 44 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2005, 89-110. doi: 10.1007/3-540-26444-2_4. Google Scholar

[15]

W. EP. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[16]

J. Geng, $W^{1,p}$ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains, Adv. Math., 229 (2012), 2427-2448. doi: 10.1016/j.aim.2012.01.004. Google Scholar

[17]

J. GengZ. Shen and L. Song, Uniform $W^{1,p}$ estimates for systems of linear elasticity in a periodic medium, J. Funct. Anal., 262 (2012), 1742-1758. doi: 10.1016/j.jfa.2011.11.023. Google Scholar

[18]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1983. Google Scholar

[19]

B. GustafssonJ. Mossino and C. Picard, $H$-convergence for stratified structures with high conductivity, Adv. Math. Sci. Appl., 4 (1994), 265-284. Google Scholar

[20]

B. Heron and J. Mossino, $H$-convergence and regular limits for stratified media with low and high conductivities, Appl. Anal., 57 (1995), 271-308. doi: 10.1080/00036819508840352. Google Scholar

[21]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. Google Scholar

[22]

C. E. KenigF. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal., 203 (2012), 1009-1036. doi: 10.1007/s00205-011-0469-0. Google Scholar

[23]

C. E. KenigF. Lin and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc., 26 (2013), 901-937. doi: 10.1090/S0894-0347-2013-00769-9. Google Scholar

[24]

C. E. Kenig and Z. Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math., 64 (2011), 1-44. doi: 10.1002/cpa.20343. Google Scholar

[25]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189-206. Google Scholar

[26]

W. NiuZ. Shen and Y. Xu, Convergence rates and interior estimates in homogenization of higher order elliptic systems, J. Funct. Anal., 274 (2018), 2356-2398. doi: 10.1016/j.jfa.2018.01.012. Google Scholar

[27]

Z. Shen, Bounds of Riesz transforms on $L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier, 55 (2005), 173-197. doi: 10.5802/aif.2094. Google Scholar

[28]

Z. Shen, The $L^p$ boundary value problems on Lipschitz domains, Adv. Math., 216 (2007), 212-254. doi: 10.1016/j.aim.2007.05.017. Google Scholar

[29]

Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653. Google Scholar

[30]

Z. Shen, Periodic Homogenization of Elliptic Systems, Advances in Partial Differential Equations, No. 269, Birkhuser Basel, 2018. doi: 10.1007/978-3-319-91214-1. Google Scholar

[31]

Z. Shen and J. Zhuge, Convergence rates in periodic homogenization of systems of elasticity, Proc. Amer. Math. Soc., 145 (2017), 1187-1202. doi: 10.1090/proc/13289. Google Scholar

[32]

S. Shkoller, An approximate homogenization scheme for nonperiodic materials, Comput. Math. Appl., 33 (1997), 15-34. doi: 10.1016/S0898-1221(97)00003-5. Google Scholar

[33]

T. A. Suslina, On the averaging of a periodic elliptic operator in a strip, Algebra i Analiz, 16 (2004), 269-292. doi: 10.1090/S1061-0022-04-00849-0. Google Scholar

[34]

T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: $L^2$-operator error estimates, Mathematika, 59 (2013), 463-476. doi: 10.1112/S0025579312001131. Google Scholar

[35]

T. A. Suslina, Homogenization of the Neumann problem for elliptic systems with periodic coefficients, SIAM J. Math. Anal., 45 (2013), 3453-3493. doi: 10.1137/120901921. Google Scholar

[36]

D. TsalisT. BaxevanisG. Chatzigeorgiou and N. Charalambakis, Homogenization of elastoplastic composites with generalized periodicity in the microstructure, International Journal of Plasticity, 51 (2013), 161-187. doi: 10.1016/j.ijplas.2013.05.006. Google Scholar

[37]

D. TsalisG. Chatzigeorgiou and N. Charalambakis, Homogenization of structures with generalized periodicity, Composites Part B: Engineering, 43 (2012), 2495-2512. doi: 10.1016/j.compositesb.2012.01.054. Google Scholar

[38]

F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math., 67 (1945), 83-93. doi: 10.2307/2371917. Google Scholar

[39]

Q. Xu, Convergence rates and $W^{1,p}$ estimates in homogenization theory of Stokes systems in Lipschitz domains, J. Differential Equations, 263 (2017), 398-450. doi: 10.1016/j.jde.2017.02.040. Google Scholar

[40]

Y. Xu and W. Niu, Convergence rates in almost-periodic homogenization of higher-order elliptic systems, preprint, arXiv: 1712.01744, 1-44.Google Scholar

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