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

2024/09/14 by Sungwon Cho, Cho, Sungwon, Fang, Junyuan +2
Mathematics · #35B05 #35B45 #35B65 #35K10 #35K65 #35K67 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2409.09437

openalex publication_date 2024/09/14 · openalex created_date 2024/10/24 · openalex updated_date 2026/07/28

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+(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, Hölder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper.

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