December  2016, 36(12): 6689-6714. doi: 10.3934/dcds.2016091

Calderón-Zygmund estimate for homogenization of parabolic systems

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, South Korea

2. 

Center for Mathematical Analysis and Computation (CMAC), Yonsei University, Seoul 03722, South Korea

Received  January 2016 Revised  July 2016 Published  October 2016

We establish a global Calderón-Zygmund estimate for homogenization of a parabolic system in divergence form with discontinuous coefficients in a bounded nonsmooth domain under the assumptions that the coefficients have small BMO seminorms and the boundary of the domain is $\delta$-flat for some $\delta>0$ depending on the given data.
Citation: Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
References:
[1]

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

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, AMS Chelsea Publishing, (2011). doi: 10.1090/chel/374.

[3]

S. Byun and L. Wang, Parabolic equations in Reifenberg domains,, Arch. Ration. Mech. Anal., 176 (2005), 271. doi: 10.1007/s00205-005-0357-6.

[4]

S. Byun and S. Ryu, Global estimates in Orlicz spaces for the gradient of solutions to parabolic systems,, Proc. Amer. Math. Soc., 138 (2010), 641. doi: 10.1090/S0002-9939-09-10094-1.

[5]

V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary,, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21. doi: 10.1007/s00030-009-0038-5.

[6]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations,, American Mathematical Society Colloquium Publications, (1995). doi: 10.1090/coll/043.

[7]

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

[8]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext. Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2.

[9]

J. Geng and Z. Shen, Uniform regularity estimates in parabolic homogenization,, Indiana Univ. Math. J., 64 (2015), 697. doi: 10.1512/iumj.2015.64.5503.

[10]

E. R. Reifenberg, Solution of the Plateau Problem for m -dimensional surfaces of varying topological type,, Acta Math., 104 (1960), 1. doi: 10.1007/BF02547186.

[11]

T. Toro, Doubling and flatness: geometry of measures,, Notices Amer. Math. Soc., 44 (1997), 1087.

[12]

L. Wang, A geometric approach to the Calderón-Zygmund estimates,, Acta Math. Sin. (Engl. Ser.), 19 (2003), 381. doi: 10.1007/s10114-003-0264-4.

show all references

References:
[1]

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

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, AMS Chelsea Publishing, (2011). doi: 10.1090/chel/374.

[3]

S. Byun and L. Wang, Parabolic equations in Reifenberg domains,, Arch. Ration. Mech. Anal., 176 (2005), 271. doi: 10.1007/s00205-005-0357-6.

[4]

S. Byun and S. Ryu, Global estimates in Orlicz spaces for the gradient of solutions to parabolic systems,, Proc. Amer. Math. Soc., 138 (2010), 641. doi: 10.1090/S0002-9939-09-10094-1.

[5]

V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary,, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21. doi: 10.1007/s00030-009-0038-5.

[6]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations,, American Mathematical Society Colloquium Publications, (1995). doi: 10.1090/coll/043.

[7]

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

[8]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext. Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2.

[9]

J. Geng and Z. Shen, Uniform regularity estimates in parabolic homogenization,, Indiana Univ. Math. J., 64 (2015), 697. doi: 10.1512/iumj.2015.64.5503.

[10]

E. R. Reifenberg, Solution of the Plateau Problem for m -dimensional surfaces of varying topological type,, Acta Math., 104 (1960), 1. doi: 10.1007/BF02547186.

[11]

T. Toro, Doubling and flatness: geometry of measures,, Notices Amer. Math. Soc., 44 (1997), 1087.

[12]

L. Wang, A geometric approach to the Calderón-Zygmund estimates,, Acta Math. Sin. (Engl. Ser.), 19 (2003), 381. doi: 10.1007/s10114-003-0264-4.

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