# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 803-847. doi: 10.1002/cpa.3160400607. [2] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374. [3] S. Byun and L. Wang, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176 (2005), 271-301. 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), 641-653. 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), 21-54. doi: 10.1007/s00030-009-0038-5. [6] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 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), 1-21. 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, New York, 1993. xvi+387 pp. 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), 697-733. 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), 1-92. doi: 10.1007/BF02547186. [11] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094. [12] L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.), 19 (2003), 381-396. doi: 10.1007/s10114-003-0264-4.

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##### References:
 [1] 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. [2] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374. [3] S. Byun and L. Wang, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176 (2005), 271-301. 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), 641-653. 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), 21-54. doi: 10.1007/s00030-009-0038-5. [6] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 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), 1-21. 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, New York, 1993. xvi+387 pp. 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), 697-733. 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), 1-92. doi: 10.1007/BF02547186. [11] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094. [12] L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.), 19 (2003), 381-396. doi: 10.1007/s10114-003-0264-4.
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