Marcinkiewicz regularity for singular parabolic $p$-Laplace type equations with measure data

TL;DR

This paper establishes Marcinkiewicz space regularity estimates for the spatial gradient of solutions to singular parabolic p-Laplace type equations with measure data, under certain density conditions, extending understanding of solution regularity in this context.

Contribution

It introduces new regularity estimates for solutions of singular parabolic equations with measure data, specifically in Marcinkiewicz spaces, for the first time in this setting.

Findings

Gradient of solutions belongs to Marcinkiewicz spaces under density conditions

Regularity results cover the singular range of p between 2n/(n+1) and 2 - 1/(n+1)

Provides a framework for analyzing measure data in parabolic p-Laplace equations

Abstract

We consider quasilinear parabolic equations with measurable coefficients when the right-hand side is a signed Radon measure with finite total mass, having $p$-Laplace type: $$u_t - \textrm{div} \, \mathbf{a}(Du,x,t) = \mu \quad \textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R}.$$ In the singular range $\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$, we establish regularity estimates for the spatial gradient of solutions in the Marcinkiewicz spaces, under a suitable density condition of the right-hand side measure.