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

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

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

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

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