Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains

We establish the natural Calderon-Zygmund theory for a nonlinear parabolic equation of p-Laplacian type in divergence form, ut - diva(Du, x, t) = div(vertical bar F vertical bar p-2 F) in Omega(T,) (0.1) by essentially proving that IF vertical bar F vertical bar(p)/ epsilon L-q (Omega(T)) double right arrow vertical bar Du vertical bar(p) epsilon L-q (Omega(T)) for every q epsilon [1,infinity). The equation under consideration is of general type and not necessarily of variation form, the involved nonlinearity a = a(xi, x, t) is assumed to have a small BMO seminorm with respect to (x, t)-variables and the lateral boundary 852 of the domain is assumed to be delta-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. (C) 2013 Elsevier Inc. All rights reserved.