Revista Matemática Iberoamericana
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Potential estimates and gradient boundedness for nonlinear parabolic systems
Tuomo Kuusi (1) and Giuseppe Mingione (2)
(1) Institute of Mathematics, Aalto University, P.O. Box 11100, 00076, AALTO, FINLAND(2) Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze, 53/A, 43124, PARMA, ITALY
We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary $p$-Laplacean system $$ u_t - \operatorname{div} (|Du|^{p-2}Du)=V(x,t) , $$ and provide $L^\infty$-bounds for the spatial gradient of solutions $Du$ via nonlinear potentials of the right hand side datum $V$. Such estimates are related to those obtained by Kilpeläinen and Malý [22] in the elliptic case. In turn, the potential estimates found imply optimal conditions for the boundedness of $Du$ in terms of borderline rearrangement invariant function spaces of Lorentz type. In particular, we prove that if $V\in L(n+2,1)$ then $Du \in L^\infty_{\mathrm{loc}}$, where $n$ is the space dimension, and this gives the borderline case of a result of DiBenedetto [5]; a significant point is that the condition $V \in L(n+2,1)$ is independent of $p$. Moreover, we find explicit forms of local a priori estimates extending those from [5] valid for the homogeneous case $V \equiv 0$.
Keywords: Nonlinear potentials, $p$-Laplacean, parabolic equations