Digital Object Identifier (DOI) 10.1007/s00205-011-0446-7 Arch. Rational Mech. Anal. 203 (2012) 189–216 Global Estimates for Quasilinear Elliptic Equations on Reifenberg Flat Domains Tadele Mengesha & Nguyen Cong Phuc Communicated by G. Dal Maso Abstract New global regularity estimates are obtained for solutions to a class of quasi- linear elliptic boundary value problems. The coefficients are assumed to have small BMO seminorms, and the boundary of the domain is sufficiently flat in the sense of Reifenberg. The main regularity estimates obtained are in weighted Lorentz spaces. Other regularity results in Lorentz–Morrey, Morrey, and Hölder spaces are shown to follow from the main estimates. 1. Introduction We study global regularity of solutions to quasilinear elliptic boundary value problems of the form  div (( A(x )∇u ·∇u ) p-2 2 A(x )∇u ) = div (|f | p-2 f ) in , u = 0 on ∂, (1.1) over a bounded domain  ⊂ R n . Function f is a given vector valued function at least in L p (, R n ), 1 < p < ∞. The coefficient matrix A(x ) is a symmetric matrix with measurable entries and satisfies the uniform ellipticity condition  1 |ξ | 2 ≦ A(x )ξ · ξ ≦  2 |ξ | 2 (1.2) for all ξ ∈ R n and almost every x ∈ , with  1 ≦  2 positive constants. A solution to equation (1.1) is understood in the standard weak sense, that is, u ∈ W 1, p 0 () is a weak solution of (1.1) if Tadele Mengesha’s research was done while the author was on leave of absence from Coastal Carolina University (CCU). The author acknowledges the support of CCU. The research is also supported by NSF grant DMS-0406374. Nguyen Cong Phuc was supported in part by NSF grant DMS-0901083.