On global $L^q$ estimates for systems with $p$-growth in rough domains

TL;DR

This paper establishes new global $L^q$ gradient estimates for nonlinear parabolic systems with $p$-growth in rough domains, providing a shorter proof of the nonlinear Calderón-Zygmund theory and extending results to the singular case.

Contribution

It offers a novel, shorter proof of global regularity results for $p$-Laplacian type systems and extends the Calderón-Zygmund theory to rough domains with no scaling deficit.

Findings

Bounds on the gradient in Lebesgue spaces with large exponents

A new, shorter proof of the global nonlinear Calderón-Zygmund theory

Extension of estimates to the singular case with no scaling deficit

Abstract

We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calder\'on-Zygmund theory for such systems. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integrability results. Technically interesting is the fact that our parabolic estimates have no scaling deficit with respect to the leading order term. Moreover, in the singular case, $p\in(\frac{2n}{n+2},2]$, any scaling deficit can be omitted.