Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains

We establish the natural Calderón-Zygmund theory for a nonlinear parabolic equation of p-Laplacian type in divergence form,. (0.1)ut-diva(Du,x,t)=div(|F|p-2F)in ΩT, by essentially proving that. (0.2)|F|p∈Lq(ΩT)⇒|Du|p∈Lq(ΩT), for every q∈. [1, ∞). The equation under consideration is of general type and not necessarily of variation form, the involved nonlinearity a= a(ξ, x, t) is assumed to have a small BMO semi-norm with respect to (x, t)-variables and the lateral boundary ∂. Ω of the domain is assumed to be δ-Reifenberg flat. As a consequence, we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory, but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property. We also find an optimal regularity estimate in Orlicz-Sobolev spaces for such nonlinear parabolic problems.