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Prescribing the $ Q' $-curvature in three dimension
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 |
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.
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Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
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S. N. Armstrong and Z. Shen,
Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923.
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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.
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M. Avellaneda and F. Lin,
Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.
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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. |
[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. |
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M. Briane, Homogénéisation de Matériaux Fibrés et Multi-couches, PhD Thesis, University Paris 6, Paris, 1990. |
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M. Briane,
Three models of nonperiodic fibrous materials obtained by homogenization, RAIRO Modél. Math. Anal. Numér., 27 (1993), 759-775.
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[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.
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[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.
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[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.
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|
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$H$-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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