Harmonic analysis and PDE seminar(10.27)


Harmonic analysis and PDE seminar


10:00am-12:00pm, October 27,  2015

Room 1124, New Main Building


Talk 1: 10:00-11:00am


Speaker: Professor Yuguang Shi, Beijing University


Title: A new rigidity result of conformally compact  Einstein manifolds

Abstract: A conformally compact Riemannian manifolds is a natural
generalization of hyperbolic space. In recent years there are growing
interests in the study of this geometric object  both from mathematics
and physics. In the first part of this talk, I will give a survey on
some geometric problems in this direction, including intrinsic
characterization of conformally compact manifolds, various rigidity
results on conformally compact Einstein manifolds, and then I will
mention a new rigidity result of conformally compact Einstein
manifolds obtained in my recent joint work with Li Gang and Qing Jie.

Talk 2: 

题目: Global regularities to semi-linear elliptic and parabolic equations with partially VMO coefficients

报告人: 郑神州

单位: 北京交通大学数学系

摘要: We prove globally $L^p$ estimates of the weak derivatives to Dirichlet's problem of semilinear elliptic equations and Cauchy-Dirichlet's problem of  semilinear parabolic equations of divergence form with partially VMO principle coefficients, respectively. The leading coefficients $a^{ij}(x)$ in the elliptic setting are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, and $a^{ij}(t,x)$ with $ij>1$ in the parabolic setting are assumed to only measurable in $(t, x_1)$ and having small BMO semi-norms in the other variables except that $a^{11}(t,x)$ is measurable in $t$ and has a small BMO semi-norm in the other variables. Based on their $L^p$ estimates, we establish the gradient estimates in mixed norms and in Morrey spaces to the weak solutions of corresponding semi-linear elliptic and parabolic problems, respectively.