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: |
[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.![]() ![]() ![]() |