Global Calderón–Zygmund theory for nonlinear elliptic obstacle problems with asymptotically regular nonlinearities

We study a nonlinear elliptic problem with an irregular obstacle in a bounded nonsmooth domain when the nonlinearity is merely asymptotically regular. We find an optimal regularity requirement on the associated nonlinearity and a minimal geometric condition on the boundary to ensure a global Calderón–Zygmund estimate for such an asymptotically regular obstacle problem. We assume that the associated nonlinearity has a small BMO and the boundary is sufficiently flat in the Reifenberg sense.