Optimal regularity estimates for general nonlinear parabolic equations

We develop a global Calderón-Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution $u$ and its spatial gradient $Du$ in a nonsmooth domain. The nonlinearity behaves as the parabolic $p$-Laplacian in $Du;$ its discontinuity with respect to the independent variables is measured in terms of small-BMO, while only Hölder continuity is required with respect to $u$ and the underlying domain is assumed to be $delta$-Reifenberg flat. We introduce and employ essentially a new concept of the intrinsic parabolic maximal function in order to overcome the main difficulties stemming from both the parabolic scaling deficiency and the nonlinearity of $u$-variable of such a very general parabolic operator, obtaining optimal $L^q$-estimates for the spatial gradient under a minimal geometric condition on the domain.