Speaker：Xushan Tu (The Hong Kong University of Science and Technology)

Time：2025-11-26 16:30

Location：Conference Room S204 at Experiment Building at Haiyun Campus

Abstract:

In this talk, we study a Monge-Amp\`ere obstacle problem, which was initially studied by Savin, and discuss related topics. First, we develop regularity theory for $\det D^2u=u^q$, establish Liouville theorems for its entire solutions. Next, we develop a variational framework for Aleksandrov estimates of convex solutions, identifying both isolated singularity problems and obstacle problems as its extremizer, and establish refined Alexandrov estimates under suitable assumptions. Furthermore, we present a global version of these improved Aleksandrov estimates and generalize a theorem of Caffarelli and Li. Finally, we may discuss the Monge-Amp\`ere equations with multiple isolated singularities, with particular emphasis on how obstacle problems influence their structural behavior. This is joint work with Tianling Jin and Jingang Xiong.