January  2018, 38(1): 75-90. doi: 10.3934/dcds.2018004

Regularity of elliptic systems in divergence form with directional homogenization

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA

* Corresponding author

Received  March 2016 Revised  July 2017 Published  September 2017

Fund Project: This research is supported by NSFC grant 11671316.

In this paper, we study regularity of solutions of elliptic systems in divergence form with directional homogenization. Here directional homogenization means that the coefficients of equations are rapidly oscillating only in some directions. We will investigate the different regularity of solutions on directions with homogenization and without homogenization. Actually, we obtain uniform interior $W^{1, p}$ estimates in all directions and uniform interior $C^{1, γ}$ estimates in the directions without homogenization.

Citation: Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004
References:
[1]

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

[2]

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

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[4]

M. SH. Birman and M. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Anal. Math., 83 (2001), 337-391.  doi: 10.1007/BF02790267.  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. Meth. Appl. Sci., 34 (2011), 1075-1096.  doi: 10.1002/mma.1424.  Google Scholar

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M. ChipotD. Kinderlehrer and G. V. Caffarelli, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96.  doi: 10.1007/BF00251414.  Google Scholar

[7]

H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Rational Mech. Anal., 205 (2012), 119-149.  doi: 10.1007/s00205-012-0501-z.  Google Scholar

[8]

H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500.  doi: 10.1007/s00526-010-0348-9.  Google Scholar

[9]

H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations: A revisit, preprint, arXiv: 1502. 00886v1 (2015). Google Scholar

[10]

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

[11]

C. E. KenigF. H. Lin and Z. W. 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

[12]

Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.  Google Scholar

[13]

T. A. Suslina, On homogenization for a periodic elliptic operator in a strip, St. Petersburg. Math. J., 16 (2004), 237-257.  doi: 10.1090/S1061-0022-04-00849-0.  Google Scholar

[14]

K. Yoshitomi, Band gap of the spectrum in periodically curved quantum waveduides, J. Differential Equations, 142 (1998), 123-166.  doi: 10.1006/jdeq.1997.3337.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures North-Holland Publ, 1978.  Google Scholar

[4]

M. SH. Birman and M. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Anal. Math., 83 (2001), 337-391.  doi: 10.1007/BF02790267.  Google Scholar

[5]

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

[6]

M. ChipotD. Kinderlehrer and G. V. Caffarelli, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96.  doi: 10.1007/BF00251414.  Google Scholar

[7]

H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Rational Mech. Anal., 205 (2012), 119-149.  doi: 10.1007/s00205-012-0501-z.  Google Scholar

[8]

H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500.  doi: 10.1007/s00526-010-0348-9.  Google Scholar

[9]

H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations: A revisit, preprint, arXiv: 1502. 00886v1 (2015). Google Scholar

[10]

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

[11]

C. E. KenigF. H. Lin and Z. W. 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

[12]

Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.  Google Scholar

[13]

T. A. Suslina, On homogenization for a periodic elliptic operator in a strip, St. Petersburg. Math. J., 16 (2004), 237-257.  doi: 10.1090/S1061-0022-04-00849-0.  Google Scholar

[14]

K. Yoshitomi, Band gap of the spectrum in periodically curved quantum waveduides, J. Differential Equations, 142 (1998), 123-166.  doi: 10.1006/jdeq.1997.3337.  Google Scholar

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