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：熊革 教授（同济大学）
题 目：The slicing problem by Bourgain and the extremal sections of convex bodies
时 间：2023.07.19（星期三）9:00-10:00
地 点：数学院南楼N802
摘 要：In this talk, I will first introduce the famous unsolved slicing problem by Bourgain in details. Then I will address our recent work on the extremal sections of convex bodies. Bounds for the volume of sections of convex bodies which are in the Lp John ellipsoid positions are established. Specifically, when the convex bodies are in the LYZ ellipsoid position, we construct a family of Hanner polytopes, which indeed attain the sharp bounds. This talk is based on the joint work with Xinbao Lu and Jiangyan Tao.
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报告人：徐露 教授（湖南大学）
题 目：A stronger constant rank theorem
时 间：2023.07.19（星期三）10:00-11:00
地 点：数学院南楼N802
摘 要：We will prove a stronger version of constant rank theorem for convex solutions to the following general semilinear elliptic equation $\Delta u=G(u)$ under certain conditions. As an application, we can show concave solutions to the Liouville equation are one-dimensional if the 2-hessian of any concave solution has a local minimum.
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报告人：韦韡 研究员（南京大学）
题 目：The $\sigma_{2}$-curvature equation on a compact manifold with boundary
时 间：2023.07.19（星期三）11:00-12:00
地 点：数学院南楼N802
摘 要：We first establish local $C^2$ estimates of solutions to the $\sigma_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the background metric has nonnegative mean curvature on totally non-umbilic boundary, for dimensions three and four we prove the existence of a conformal metric with a prescribed positive $\sigma_2$-curvature function and a prescribed nonnegative boundary mean curvature function. The local estimates play an important role in blow up analysis in the latter existence result.
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