Symposium on Partial Differential Equations 2017

Symposium on Partial Differential Equations 2017

Content

Time: 2017.6.26

A.M

Chair: Daomin Cao

9:00-10:00

Speaker:L.Caffarelli  (University of Texas, USA)

Title: Non local flows

Abstract: We will describe the evolution of methods to show regularity for diffusion processes: the quasi geostrophic equation, non local porous media, memory effects.

 

10:10-11:10

Speaker: Yu Yuan(University of Washington, Seattle)

Title: Asymptotic behavior of solutions to Hessian equations over exterior domains

Abstract: We present a unified approach to quadratic asymptote of solutions to a class of fully nonlinear elliptic equations over exterior domains, including Monge- Ampere equations  ( previously known ), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. This is joint work with Dongsheng Li and Zhisu Li.

 

P.M

Chair: Feimin Huang

 

14:30-15:30

Speaker: Jingang Xiong (Beijing Normal University, P.R.China)

Title: Boundary singularities for elliptic equations

Abstract: The isolated singularity problem for the Yamabe equation has been very well understood since the seminal work of Caffarelli, Gidas and Spruck. In this talk, I will present some results about boundary singularities for elliptic equations with Neumann or Dirichlet conditions. Conformal invariance is a common feature of them. The Neumann problem arises from the recent studies of fractional GJMS operators on the conformal infinity of Poincare-Einstein manifolds. The Dirichlet problem can be viewed as an analogue of Caffarelli-Gidas-Spruck on the boundary, which, however, is open. A partial result will be given. Some new idea here will be used to solve an isoperimetric problem over scalar flat conformal class. This talk partially bases on joint work with L. Caffarelli, T. Jin, O. de Queiroz, Y. Sire and L. Sun.

 

 15:40-16:40

Speaker: Xinan Ma (Chinese University of Science and Technology)

Title: Neumann boundary value problem for k-Hessian nonlinear elliptic equation

Abstract: For k-Hessian nonlinear elliptic equation, in 1985, Caffarelli- Nirenberg -Spruck obtained the existence of the classical admissible  solution with Dirichlet problem under uniformly k-1 convex domain. In 1987, Trudinger solved the Neumann problem for the k-Hessian equation when the domain is a ball, and he conjectured the existence for sufficiently smooth uniformly convex domains.  In this talk, we prove the existence of a classical admissible solution to a Neumann boundary value problem for k-Hessian equations in uniformly convex domain, and in some case even for uniformly k-1 convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we give an affirmative answer to a conjecture of Trudinger in 1987.  This is a joint work with Guohuan Qiu (Arxiv 1508.00196).