中科院数学与系统科学研究院
数学研究所
学术报告
偏微分方程研讨班
报告人： 汪铃（北京大学）
题 目：A revisit to affine Bernstein problem
时 间：2022.03.30（星期三）14:00-15:30
地 点：思源楼S817
摘 要：In this report, I'll go through the proof of affine Bernstein problem given by Trudinger and Wang \cite{TW}, and I mainly introduce the idea about how to get the all dimensional conclusions under the assumption of uniform, "strict convexity" and just mention the proof of dimension two, which is the Chern conjecture \cite{Ch}. It is because their method for dimension two can not be extended to higher dimensions, even for dimension 3 and there are also other proofs of Chern's conjecture in dimension two. In the end, it is also worthy to mention that they produced a (non-smooth) counterexample for $n\geq 10$. This lecture is a seminar report，mainly introduce the existence work.
----------------------------
报告人：黄耿耿 教授（复旦大学）
题 目：Regularity of free boundary for the Monge-Ampere obstacle problem
时 间：2022.04.13（星期三）10:00-11:00
地 点：腾讯会议：948-3602-5762
摘 要：In this talk, we talk about the regularity of the free boundary in the Monge-Amp\`ere obstacle problem
\begin{equation}\begin{split}\det D^2 v=f(y)\chi_{\{v>0\}},\quad \text{in}\quad \Omega\\v=v_0,\quad\text{on}\quad\partial\Omega.\end{split}\end{equation} Assume that $\Omega$ is a bounded convex domain in $\Bbb R^n$, and $f, v_0>0$. Then $\Gamma=\partial \{v=0\}$ is smooth if $f$ is smooth; and $\Gamma$ is analytic if $f$ is analytic. This is a joint work with Prof. Tang Lan and Prof. Wang Xu-Jia.
附件：