Harnack inequality for singular or degenerate parabolic equations in non-divergence form

TL;DR

This paper establishes a Harnack inequality for a class of singular or degenerate linear parabolic equations in non-divergence form with weights in the Muckenhoupt class, leading to regularity results and a Liouville theorem.

Contribution

It introduces a new weighted approach and smallness conditions to prove Harnack inequalities for equations with singular or degenerate coefficients in non-divergence form.

Findings

Proved Krylov-Safonov Harnack inequality under weighted smallness conditions.

Established Hölder regularity estimates for solutions.

Derived a Liouville type theorem for the class of equations.

Abstract

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additionally, a perturbation method is used and the parabolic Aleksandrov-Bakelman-Pucci type maximum principle is crucially applied to suitable barrier functions to control the solutions. As corollaries, H\"{o}lder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper.