Gradient higher integrability for singular parabolic double-phase systems

TL;DR

This paper establishes a local higher integrability property for gradients of solutions to certain singular parabolic double-phase systems, expanding understanding of their regularity under specific geometric and scaling conditions.

Contribution

It introduces a novel reverse H"older inequality in intrinsic cylinders that combines different geometries, advancing regularity theory for double-phase systems.

Findings

Proves higher integrability for gradients in specified p-range.

Develops a reverse H"older inequality in intrinsic geometries.

Identifies the impact of singular scaling deficits on q-range.

Abstract

We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of $p$-Laplace type when $\tfrac{2n}{n+2}< p\le2$. The result is based on a reverse H\"older inequality in intrinsic cylinders combining $p$-intrinsic and $(p,q)$-intrinsic geometries. A singular scaling deficits affects the range of $q$.