# American Institute of Mathematical Sciences

September  2021, 20(9): 3143-3159. doi: 10.3934/cpaa.2021100

## $W^{1,p}$ estimates for elliptic systems on composite material with almost partially BMO coefficients

 1 School of Mathematical Sciences, Peking Uinversity, Beijing 100871, China 2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author

Received  April 2018 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: The first author is supported by NSF grant 8206400077

In this paper, we establish uniform $W^{1,p}$ estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global $W^{1,p}$ regularity under the assumption that the coefficients are almost $(\delta,R)$-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains.

Citation: Caiyan Li, Dongsheng Li. $W^{1,p}$ estimates for elliptic systems on composite material with almost partially BMO coefficients. Communications on Pure &amp; Applied Analysis, 2021, 20 (9) : 3143-3159. doi: 10.3934/cpaa.2021100
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##### References:
 [1] S. Byun, S. Ryu and L. Wang, Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains, Manuscripta Math., 133 (2010), 225-245.  doi: 10.1007/s00229-010-0373-1.  Google Scholar [2] S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037.  Google Scholar [3] L. 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.  Google Scholar [4] M. Chipot, D. Kinderlehrer and G. V. Caffarelli, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96.  doi: 10.1007/BF00251414.  Google Scholar [5] Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients (English summary), Arch. Ration. Mech. Anal., 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar [6] Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.  Google Scholar [7] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, (1993). doi: 10.1515/9781400883929.  Google Scholar [8] K. Um, Elliptic equations with singular BMO coefficients in Reifenberg domains, J. Differ. Equ., 253 (2012), 2993-3015.  doi: 10.1016/j.jde.2012.08.016.  Google Scholar
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