Elliptic equations with measurable nonlinearities in nonsmooth domains

We study elliptic equations with measurable nonlinearities in nonsmooth domains. We establish an optimal global W1,p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small BMO semi-norm in the other variables, while the underlying domain is sufficiently flat in the Reifenberg sense that the boundary of the domain is locally trapped between two narrow strips.