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arXiv · 2409.09437

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

Sungwon Cho·Junyuan Fang·Tuoc Phan
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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, 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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Sungwon Cho, Junyuan Fang, Tuoc Phan. 2024-10-10. Harnack inequality for singular or degenerate parabolic equations in non-divergence form. https://arxiv.org/abs/2409.09437

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