Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains (Q393033)

The interesting paper under review deals with Calderón-Zygmund theory for divergence form nonlinear parabolic equations of \(p\)-Laplacian type NEWLINE\[NEWLINE u_t - \text{div\,} \mathbf{a}(Du,x,t)= \text{div\,}(|F|^{p-2}F)\quad \text{in}\;\Omega_T NEWLINE\]NEWLINE where \(\Omega_T\) is the cylinder \(\Omega\times(0,T)\) and \(\Omega\subset{\mathbb R}^n\) is a bounded domain with Reifenberg flat boundary \(\partial\Omega\) and \(p\in\left({{2n}\over{n+2}},\infty\right).\)NEWLINENEWLINEThe nonlinearity involved \(\mathbf{a}(\xi,x,t)\) is supposed to be elliptic and of Carathéodory type, and to have small-BMO seminorms with respect to the variables \((x,t).\)NEWLINENEWLINEThe main result of the paper establishes a Calderón-Zygmund type regularity for the weak solutions of the problem considered. Namely, it is proved that NEWLINE\[NEWLINE |F|^p\in L^q(\Omega_T)\;\Longrightarrow\;|Du|^p\in L^q(\Omega_T)\quad \forall q\in[1,\infty). NEWLINE\]NEWLINE As consequence, the authors not only relax the known smoothness requirements on the nonlinearity, but also extend the local regularity theory to a global one in the case of nonsmooth domains with boundaries of fractal type. Moreover, optimal regularity estimates are derived for such nonlinear parabolic problems in the framework of the Sobolev-Orlicz spaces.