Borderline gradient continuity for nonlinear parabolic systems

Abstract

We consider the evolutionary \(p\)-Laplacean system $$\begin{aligned} \partial _t u-\triangle _p u=F,\qquad p > \frac{2n}{n+2} \end{aligned}$$ in cylindrical domains of \( \mathbb R^{n}\times \mathbb R\), and prove the continuity of the spatial gradient \(Du\) under the Lorentz space assumption \(F\in L(n+2,1)\). When \(F\) is time independent the condition improves in \(F \in L(n,1)\). This is the limiting case of a result of DiBenedetto claiming that \(Du\) is Holder continuous when \(F \in L^{q}\) for \(q>n+2\). At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system \(\triangle u \in L(n,1)\) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.