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

Regularity of elliptic systems in divergence form with directional homogenization

Rong Dong 1,, , Dongsheng Li 1, and  Lihe Wang 2,

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.

Keywords: Regularity, directional homogenization, elliptic systems in divergence form.
Mathematics Subject Classification: 35B27, 35B65, 35J47.
Citation: Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.

[5]

R. Bunoiu, G. 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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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.  doi: 10.1090/S0894-0347-2013-00769-9.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[5]

R. Bunoiu, G. 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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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.  doi: 10.1090/S0894-0347-2013-00769-9.

[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.

[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.

[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.

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