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

Borderline gradient continuity for nonlinear parabolic systems

created by mingione on 20 Nov 2013
modified on 12 Nov 2014


Published Paper

Inserted: 20 nov 2013
Last Updated: 12 nov 2014

Journal: Math. Ann.
Volume: 360
Number: 3-4
Pages: 937–993
Year: 2014


We consider the evolutionary $p$-Laplacean system \[\partial_t u-\triangle_p u=F ,\qquad \quad p > \frac{2n}{n+2} \] 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 H\"older 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


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