Periodic homogenization of elliptic systems with stratified structure

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

Periodic homogenization of elliptic systems with stratified structure

Yao Xu ^1, and Weisheng Niu ^2,,

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.

Keywords: Homogenization, periodic coefficients, stratified structure, convergence rates, uniform regularity estimates.

Mathematics Subject Classification: Primary: 35B27; Secondary: 35J47.

Citation: Yao Xu, Weisheng Niu. Periodic homogenization of elliptic systems with stratified structure. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2295-2323. doi: 10.3934/dcds.2019097

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	M. Briane, Homogénéisation de Matériaux Fibrés et Multi-couches, PhD Thesis, University Paris 6, Paris, 1990.
[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.
[11]	R. Bunoiu, G. 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.
[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.
[13]	R. Dong, D. 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.
[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.
[15]	W. E, P. 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.
[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.
[17]	J. Geng, Z. 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.
[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.
[19]	B. Gustafsson, J. Mossino and C. Picard, $H$-convergence for stratified structures with high conductivity, Adv. Math. Sci. Appl., 4 (1994), 265-284.
[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.
[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.
[22]	C. E. Kenig, F. 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.
[23]	C. E. Kenig, F. 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.
[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.
[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.
[26]	W. Niu, Z. 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.
[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.
[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.
[29]	Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653.
[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.
[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.
[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.
[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.
[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.
[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.
[36]	D. Tsalis, T. Baxevanis, G. 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.
[37]	D. Tsalis, G. 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.
[38]	F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math., 67 (1945), 83-93. doi: 10.2307/2371917.
[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.
[40]	Y. Xu and W. Niu, Convergence rates in almost-periodic homogenization of higher-order elliptic systems, preprint, arXiv: 1712.01744, 1-44.

show all references

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	M. Briane, Homogénéisation de Matériaux Fibrés et Multi-couches, PhD Thesis, University Paris 6, Paris, 1990.
[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.
[11]	R. Bunoiu, G. 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.
[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.
[13]	R. Dong, D. 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.
[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.
[15]	W. E, P. 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.
[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.
[17]	J. Geng, Z. 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.
[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.
[19]	B. Gustafsson, J. Mossino and C. Picard, $H$-convergence for stratified structures with high conductivity, Adv. Math. Sci. Appl., 4 (1994), 265-284.
[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.
[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.
[22]	C. E. Kenig, F. 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.
[23]	C. E. Kenig, F. 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.
[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.
[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.
[26]	W. Niu, Z. 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.
[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.
[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.
[29]	Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653.
[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.
[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.
[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.
[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.
[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.
[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.
[36]	D. Tsalis, T. Baxevanis, G. 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.
[37]	D. Tsalis, G. 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.
[38]	F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math., 67 (1945), 83-93. doi: 10.2307/2371917.
[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.
[40]	Y. Xu and W. Niu, Convergence rates in almost-periodic homogenization of higher-order elliptic systems, preprint, arXiv: 1712.01744, 1-44.

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