Global Calder\`on & Zygmund theory for nonlinear parabolic systems

Abstract

We establish a global Calder\'on & Zygmund theory for solutions of a huge class of nonlinear parabolic systems whose model is the inhomogeneous parabolic $p$-Laplacian system \begin{equation*} \left\{\begin{array}{cc} \partial_t u - \Div (|Du|^{p-2}Du) = \Div (|F|^{p-2}F) &\mbox{in $\Omega_T:=\Omega\times(0,T)$} \\[5pt] u=g &\mbox{on $\partial\Omega\times(0,T)\cup \bar\Omega\times\{0\}$} \end{array}\right. \end{equation*} with given functions $F$ and $g$. Our main result states that the spatial gradient of the solution is as integrable as the data $F$ and $g$ up to the lateral boundary of $\Omega_T$, i.e. \begin{equation*} F,Dg\in L^q(\Omega_T),\ \ \partial_t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega_T) \quad\Rightarrow \quad Du\in L^q(\Omega\times(\delta,T)) \end{equation*} for any $q>p$ and $\delta\in(0,T)$, together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.