A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,A(x,t,Du)=0 \quad\text{in }E_T\subset \R^N\times\R, $$ 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 $L^p_{loc}$ to $L^{p(1+\varepsilon)}_{loc}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms.
Regularity theory for sub-critical p-parabolic systems with measurable coefficients
Gianazza, Ugo
;
2026-01-01
Abstract
A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,A(x,t,Du)=0 \quad\text{in }E_T\subset \R^N\times\R, $$ 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 $L^p_{loc}$ to $L^{p(1+\varepsilon)}_{loc}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms.File in questo prodotto:
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