Square function estimates for the evolutionary p-Laplace equation

TL;DR

This paper establishes new local square function and Carleson measure estimates for solutions to the evolutionary p-Laplace equation, extending tools used in classical PDEs to a more complex, non-homogeneous setting.

Contribution

It introduces novel square function estimates for the evolutionary p-Laplace equation, with initial applications to parabolic uniform rectifiability and boundary behavior analysis.

Findings

Square function estimates are proven for solutions to the evolutionary p-Laplace equation.

Applications to parabolic uniform rectifiability and boundary behavior are demonstrated.

The results extend classical PDE tools to non-homogeneous, evolutionary equations.

Abstract

We prove novel (local) square function/Carleson measure estimates for non-negative solutions to the evolutionary $p$-Laplace equation in the complement of parabolic Ahlfors-David regular sets. In the case of the heat equation, the Laplace equation as well as the $p$-Laplace equation, the corresponding square function estimates have proven fundamental in symmetry and inverse/free boundary type problems, and in particular in the study of (parabolic) uniform rectifiability. Though the implications of the square function estimates are less clear for the evolutionary $p$-Laplace equation, mainly due its lack of homogeneity, we give some initial applications to parabolic uniform rectifiability, boundary behaviour and Fatou type theorems for $\nabla_Xu$.