Calculus of Variations and Geometric Measure Theory
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T. Kuusi - G. Mingione

Potential estimates and gradient boundedness for nonlinear parabolic systems

created by mingione on 28 Sep 2012
modified on 29 Sep 2012


Published Paper

Inserted: 28 sep 2012
Last Updated: 29 sep 2012

Journal: Revista Matematica Iberoamericana
Volume: 28
Pages: 535-576
Year: 2012


We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary $p$-Laplacean system \[ u_t -div\ (|Du|^{p-2}Du)=V(x,t)\,, \] 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 Kilpel\"ainen \& Mal\'y \cite{KM} 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 $Du \in L^\infty$ locally, where $n$ is the space dimension, and this gives the borderline case of a result of DiBenedetto \cite{DB2}; 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 \cite{DB2} valid for the homogeneous case $V \equiv 0$.


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