Borderline gradient continuity for nonlinear parabolic systems (Q476772)

The authors consider the evolutionary \(p\)-Laplacean system \[ \partial_tu-\Delta_pu=F,\,\,\, p>\frac{2n}{n+2} \] in cylindrical domains of \(\mathbf{R}^n\times \mathbf{R}\), where \(F\) belongs to the Lorenz space \(L(n+2,1)\). Under such a hypothesis they prove the continuity of the spatial gradient \(Du\). In the special case \(F\) time independent, they prove that, to get such a result, it is enough to assume \( F\in L(n,1)\). This is the limiting case of results proved by DiBenedetto and Stein. DiBenedetto proved that \(Du\) is Hölder continuous when \(F\in L^q\) for \(q>n+2\). Stein proved that the gradient of the solutions to the elliptic linear system \(\Delta u \in L(n,1)\) is continuous. In this paper, moreover, the authors prove new potential estimates and introduce very interesting nonlinear potentials.