Optimal regularity estimates for general nonlinear parabolic equations

Abstract

We develop a global Calderon–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 Holder 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.