Very weak solutions to degenerate parabolic double-phase systems

TL;DR

This paper establishes a local self-improving property for the gradient of very weak solutions to degenerate parabolic double-phase systems, introducing a new phase analysis method and reverse H"older inequality independent of solutions.

Contribution

It introduces a novel phase analysis technique and proves a reverse H"older inequality for very weak solutions, advancing understanding of degenerate parabolic systems.

Findings

Proves a self-improving gradient property for solutions.

Develops a new phase analysis method.

Establishes a reverse H"older inequality with solution-independent constants.

Abstract

We prove a local self-improving property for the gradient of very weak solutions to degenerate parabolic double-phase systems. The result is based on a reverse H\"older inequality with constants that are independent of the solution. Delicate methods are required to avoid a self-referential argument. In particular, we develop a new phase analysis method.