Title: Harnack Inequality for Singular or Degenerate Parabolic Equations in Non-Divergence Form
Speaker: Junyuan Fang
Institution: University of Tennessee, Knoxville
Date & Time: December 26, 2024, 9:00–10:00
Location: Second Lecture Hall, 1st Floor, Mathematics Building
On-Campus Contact: Yuanyuan Nie, nieyy@jlu.edu.cn
This talk addresses a class of linear parabolic equations in non-divergence form, where the leading coefficients are measurable and exhibit singularity or degeneracy as weights belonging to the
A1+n1 class of Muckenhoupt weights. Under a smallness assumption on the weighted mean oscillation of the weight, we prove the Krylov-Safonov Harnack inequality for solutions. To establish this result, we introduce a class of generic weighted parabolic cylinders and impose a smallness condition on the weighted mean oscillation, through which several growth lemmas are derived. Additionally, a perturbation method is employed, and the parabolic Aleksandrov-Bakelman-Pucci (ABP)-type maximum principle is critically 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 presented.
This work is joint with Sungwon Cho and Tuoc Phan.
Junyuan Fang is a Ph.D. candidate at the University of Tennessee, Knoxville. He earned his master’s degree from Capital Normal University in 2020. His research focuses on the regularity theory of degenerate elliptic and parabolic equations, as well as related problems in the calculus of variations.
