Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations

TL;DR

This paper proves weak Harnack inequalities for positive supersolutions of a class of doubly degenerate parabolic equations, expanding understanding of their regularity properties.

Contribution

It introduces new weak Harnack estimates for a broad class of doubly degenerate parabolic equations, using Caccioppoli, De Giorgi, and Moser techniques.

Findings

Established weak Harnack inequalities for supersolutions

Extended regularity results to doubly degenerate equations

Applied iterative methods to prove inequalities

Abstract

We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad m+p>3.$$ Our proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.