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,equation(0.1)ut−diva(Du,x,t)=div(|F|p−2F)in ΩT, by essentially proving thatequation(0.2)|F|p∈Lq(ΩT)⇒|Du|p∈Lq(ΩT), for every q∈[1,∞)q∈[1,∞). The equation under consideration is of general type and not necessarily of variation form, the involved nonlinearity a=a(ξ,x,t)a=a(ξ,x,t) is assumed to have a small BMO semi-norm with respect to (x,t)(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.