H\"older-Zygmund Estimates for Degenerate Parabolic Systems

TL;DR

This paper establishes H"older-Zygmund regularity estimates for solutions of degenerate parabolic systems with inhomogeneous terms in BMO, extending Calderón-Zygmund theory and analyzing gradient decay for p-caloric solutions.

Contribution

It proves new regularity results for energy solutions of the p-Laplacian system with BMO right-hand side, including local estimates and gradient decay properties.

Findings

Solutions are locally in L^ abla( ext{C}^1) for p ≥ 2.

Finer properties like H"older continuity are preserved by the gradient.

New decay estimates for gradients of p-caloric solutions for p in (2n/(n+2), ∞).

Abstract

We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien system $\partial_t u-\text{div}(|D u|^{p-2}D u)=-\text{div} g$. We show in the case $p\geq 2$ that if the right hand side $g$ is locally in $L^\infty(\text{BMO})$, then $u$ is locally in $L^\infty(\mathcal{C}^1)$, where $\mathcal{C}^1$ is the 1-H\"older--Zygmund space. This is the borderline case of the Calder\'on-Zygmund theorey. We provide local quantitative estimates. We also show that finer properties of $g$ are conserved by $D u$, e.g.\ H\"older continuity. Moreover, we prove a new decay for gradients of $p$-caloric solutions for all $\frac{2n}{n+2}<p<\infty$.