Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients

TL;DR

This paper develops a quantitative regularity theory for weak solutions to sub-critical p-parabolic systems with measurable coefficients, establishing local boundedness and higher gradient integrability results.

Contribution

It provides the first sharp, scale-invariant $L^ abla$ estimates and gradient self-improvement results for such systems in the sub-critical range.

Findings

Established local boundedness from $L^{oldsymbol{ }}$ control

Proved higher integrability of the gradient $|Du|$

Results hold with proper source terms

Abstract

A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1<p\le \tfrac{2N}{N+2}$ and obtain two main results. \emph{Local boundedness:} starting from an $L^{\boldsymbol{\mathsf r}}$-control of $u$ with ${\boldsymbol{\mathsf r}}>\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\mathrm{loc}}$ 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.