Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains

We consider a double phase problem with BMO coefficient in divergence form on a bounded nonsmooth domain. The problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position, which describes a feature of strongly anisotropic materials. We obtain the global Calderón-Zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small BMO and the boundary of the domain is sufficiently flat in the Reifenberg sense.