Log‐BMO matrix weights and quasilinear elliptic equations with Orlicz growth in Reifenberg domains

Abstract We study a very general quasilinear elliptic equation with the nonlinearity with Orlicz growth subject to a degenerate or singular matrix weight on a bounded nonsmooth domain. The nonlinearity satisfies a nonstandard growth condition related to the associated Young function, and the logarithm of the matrix weight in BMO (bounded mean oscillation) is constrained by a smallness parameter which has a close relationship with the Young function. We establish a global Calderón–Zygmund estimate for the weak solution of such a degenerate or singular problem in the setting of a weighted Orlicz space under a minimal geometric assumption that the boundary of the domain is sufficiently flat in the Reifenberg sense. Our regularity result is, up to our knowledge, the first one available for divergence structure quasilinear elliptic equations with matrix weights and nonstandard growth in the literature.