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

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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: W^1,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 and 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.
[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.
[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.
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[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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