## Lectures on Geometric PDEs

**Speaker(s): ** Various Speakers

**Time: **June 20 - June 21, 2015

**Venue: ** Room 09 at Quan Zhai, BICMR

**Time**: June 20-21, 2015

**Place**: Quan9, Jingchunyuan 78 yard, BICMR

**Aim**: In this two-day workshop, there will be four lectures on topics which will be of interest to a broad range of audiences.

**Schedule**:

June 20, Saturday

9:00-11:30: Wu Jie, Volume estimates on the critical sets of elliptic equations

13:00-15:30: Xiong Jingang, Asymptotic analysis for positive solutions

June 21, Sunday

9:00-11:30: Jiang Xumin, The linearization of the complex Monge-Ampere equation and the tangential estimates

13:00-15:30: Chen Chuanqiang, The interior $C^2$ a priori estimate for the admissible solution of the $\sigma_2$-equation

**Abstracts**:

Title: The interior $C^2$ a priori estimate for admissible solutions of the $\sigma_2$-equation

Speaker: Chen Chuanqiang, Zhejiang University of Technology

Abstract: The interior $C^2$ a priori estimate for admissible solutions of $\sigma_2(D^2 u)=f(x)>0$ is a longstanding and important problem. For the two dimension, the equation is a Monge-Ampere equation, and the estimate was firstly proved by Heinz [J. Analyse Math., 1959], and a new proof was obtained by Chuanqiang Chen, Fei Han and Qianzhong Ou recently. For the three dimension and $f =1$, the equation is a special Lagrangian equation, and the estimate was obtained by Warren-Yuan [CPAM, 2009]. In this talk, we discuss the history and the development of this problem.

Title: The linearization of the complex Monge-Ampere equation and the tangential estimates

Speaker: Jiang Xumin, University of Notre Dame

Abstract: Cheng and Yau proved the existence of the complete Kahler-Einstein metrics on strongly pseudo-convex domains. Lee and Melrose studied the optimal regularity up to the boundary. In this talk, we discuss the underlying complex Monge-Ampere equations. The talk consists of two parts. First, we calculate the linearized operator under normal frames. Second, we derive the tangential estimates. Based on these two parts, we prove the boundary expansion and convergence for the complex Monge-Ampere equation.

Title: Volume estimates on the critical sets of elliptic equations

Speaker: Wu Jie (BICMR)

Abstract: In this talk, we present Naber and Valtorta’s recent work on the critical and singular sets of elliptic equations. They introduce new techniques for estimating these sets, which avoids the need of smoothness of the coefficients, and they give upper bounds for the $n-2$ dimensional Minkowski measure of these sets under merely Lipschitz assumption of the leading coefficients.

Title: Asymptotic analysis for positive solutions of nonlinear elliptic equations with an isolated singularity

Speaker: Xiong Jingang, Beijing Normal University

Abstract: Understanding asymptotic behaviors of solutions of PDEs near an isolated singularity is of the basic importance. In this lecture, we will report a classical result of Caffarelli-Gidas-Spruck [Comm Pure Appl Math, 42 (1989), 271—297] on the isolated singularity of the Yamabe equation. A proof of this result based on the moving sphere method will be presented. We will also report some refined results due to Korevaar-Mazzeo_pacard-Schoen [Invent. Math, 135 (1999), 233-272], which gives the next order expansion. Refined results are crucial to solving singular Yamabe problem by the gluing method.