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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

Caiyan Li 1,, and  Dongsheng Li 2,

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.

Keywords: W1,p estimates, discontinuous coefficients, elliptic systems, composite material.
Mathematics Subject Classification: Primary: 35D10, 35J30, 35J50; Secondary: 35L30.
Citation: Caiyan Li, Dongsheng Li. $ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3143-3159. doi: 10.3934/cpaa.2021100
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

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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

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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

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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

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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

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show all references

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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