| Abstract: |
| A quantitative regularity theory is developed for weak solutions to the parabolic system
$$
\partial_t u-\mathrm{div}\,\sfA(x,t,Du)=0
\quad\text{in }E_T\subset \R^N\times\R,
$$
which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\rm loc}$ to $L^{p(1+\varepsilon)}_{\mathrm{loc}}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms. |
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