## 非线性偏微分方程与几何分析研讨会

张 希（中国科学技术大学）   李奇睿（网投十大信誉网站）

 12月27日（周五） 9:00-9:10 开幕致辞 9:10-9:55 汪徐家 Some curvature related problems 10:00-10:45 王克磊 Regularity of transition layers in Allen-Cahn   equation 茶 歇 11:00-11:45 李海刚 Babuska   Problem in Composite Materials and its Applications 午 餐 14:00-14:45 陈世炳 Regularity   of optimal transport map between hypercubes 14:50-15:35 徐 露 Solutions   to the equations from the conformal geometry 茶 歇 15:50-16:35 陈传强 The   Neumann problems of some fully nonlinear elliptic PDEs 16:40-17:25 张德凯 On   degenerate k-Hessian equations on compact Hermitian manifolds 晚 餐

 12月28日（周六） 9:00-9:45 朱小华 Variation, polytopes   and KE metrics on Q-Fano varieties 9:50-10:35 韩小利 Stability of line bundle mean curvature flow 合 影,茶 歇 11:00-11:45 朱苗苗 Regularity   for critical elliptic systems and applications in geometric analysis 午 餐 14:00-14:45 夏 超 Symmetrization   with respect to mixed volumes 14:50-15:35 邱国寰 On   degenerate case of prescribed curvature measure problems 茶 歇 15:50-16:35 孙 俊 Variational   characterizations of invariant submanifolds in Kahler and Sasaki manifolds 16:40-17:25 熊 革 On the   applications and generalizations of the cone-volume functional 晚 宴

 12月29日（周日） 9:00-9:45 李东升 Interior   $L^p$ Estimates of Partial Differential Equations and Systems 9:50-10:35 周显潮 Scalar   curvatures in almost Hermitian geometry and some applications 茶 歇 11:00-11:45 唐 岚 Global   C^{1, \alpha} regularity for convex solutions to the   Dirichlet problems of the degenerate Monge-Ampere equations 午 餐 14:00-14:45 刘佳堃 Regularity   of free boundaries in optimal transportation 14:50-15:35 蒋飞达 Neumann   problem for degenerate Monge-Ampere equation 茶 歇 16:00-17:00 自由讨论 晚 餐

(按报告人姓名音序排列)

I will first review some of our results on upper and lower bounds of the gradients by developing an iteration technique with respect to the energy integral to overcome the essential difficulty from the lack of maximal principle for elliptic systems, then present our very recent results, improving the previous inequality results on the gradients’ estimates to asymptotics to close this problem and show the key role of the geometry of the inclusions played in such blow-up analysis.

One observation is that by a simple comparison principle we can reduce the key estimate into the corresponding estimate to a homogeneous mean curvature equation. Then by analysis the existence of homogeneous mean curvature equation, we can give a sufficient condition such that the prescribed function may touch zero somewhere.

In this talk, we will first review the fundamental properties of the cone-volume functional and its applications to the Schneider projection problem. Then, we will talk on the solved LYZ conjecture for the cone-volume functional (including our work on this conjecture) and its applications to the logarithmic Minkowski problem. Then we will report our very recent results on its generalizations and applications, including the variational formula and the extreme problem on the mixed cone-volume functional.

This talk is based on the joint work with Hu Jiaqi, Lu Xinbao and Sun Qiang.