学术报告
The 2016 Xiamen International Conference on Partial Differential Equations and Applications
编辑:发布时间:2016年06月13日

会议名称:The 2016 Xiamen International Conference on Partial Differential Equations and Applications

日期:6月15日-6月18日

地点:海韵园数学物理大楼117报告厅

 

6月15日,海韵园数学物理大楼117

8:25

逸夫楼大堂集合,统一乘车去会场

8:50-9:00

开幕式

上午9:00-12:30,学术报告,主持人,Yisong Yang

9:00-9:45

Joel SpruckJohns Hopkins UniversitySelf-shrinkers to the mean curvature flow asymptotic to an isoparametric cone

9:50-10:35

Harold Rosenberg(IMPA, Brazil)Minimal surfaces in closed and finite volume hyperbolic 3-manifolds

10:35-10:55

茶歇

上午10:55-12:30,学术报告,主持人,Xujia Wang

10:55-11:40

Paul Yang(Princeton University)Fourth order equations in conformal and CR geometry

11:45-12:30

Pengfei Guan(McGill University)Regularity of immersed  hypersurfaces in Riemannian manifolds

12:30-14:30

午餐、休息

下午14:30-16:05,学术报告,主持人,Yanyan Li

14:30-15:15

Alice S.-Y. Chang(Princeton University)Boundary type GJMS operators

15:20-16:05

Jixiang Fu(Fudan University) Some results on positivity in Kahler geometry

16:05-16:25

茶歇

下午16:25-18:00,学术报告,主持人,Gongbao Li

16:25-17:10

Yanyan Li(Rutgers University)Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

17:15-18:00

Xiaohua Zhu(Peking University) Properness of F-functional


6月16日,海韵园数学物理大楼117

8:30-8:50

科学艺术中心门口合影

8:50

科学艺术中心门口统一乘车去会场

上午9:30-10:15,学术报告,主持人,Pengfei Guan

9:30-10:15

Chang-Shou Lin(Taiwan University)Multiple Green function and Lame equation

10:15-10:35

茶歇

上午10:35-12:10,学术报告,主持人,Changshou Lin

10:35-11:20

Lihe Wang(University of Iowa)Geometric capacity of sets and regularity

11:25-12:10

Robert Gulliver(University of Minnesota) Immersed minimal surfaces,orientable or nonorientable

12:10-14:30

午餐、休息

下午14:30-18:00,学术报告,主持人,Paul Yang

14:30-15:15

Fanghua Lin(New York University)Superfluid passing an obstacle---Nucleation of vortices

15:20-16:05

Changyou Wang(Purdue University)Global estimates and bubbling analysis of approximate harmonic maps in dimension two

16:05-16:25

茶歇

下午16:25-18:00,学术报告,主持人,Fanghua Lin

16:25-17:10

Yu Yuan(University of Washington)Special Lagrangian equations

17:15-18:00

Xinan Ma(University of Science and Technology of China)Neumann boundary value problem for Hessian equations on convex domain

 


6月17日,海韵园数学物理大楼117

8:30

逸夫楼门口统一乘车去会场

上午9:00-10:35,学术报告,主持人,Robert Hardt

9:00-9:45

Richard Schoen(Stanford University and University of California Irvine)Metrics of fixed area on high genus surfaces with largest first eigenvalue

9:50-10:35

Ling Xiao(Rutgers University)Motion of level sets by general curvature flow

10:35-10:55

茶歇

上午10:55-12:30,学术报告,主持人,Lihe Wang

10:55-11:40

Xujia Wang(Australian National University)Monge-Ampere equations arising in geometric optics

11:45-12:30

Huaiyu Jian(Tsinghua University) Regularity analysis for a class of singular elliptic equation

12:30-14:30

午餐、休息

下午14:30-16:05,学术报告,主持人,Zhong Tan

14:30-15:15

Robert Hardt(Rice University)A linear isoperimetric inequality for algebraic and analytic varieties

15:20-16:05

Qing Han(University of Notre Dame) Singular solutions of Yamabe equation

 


学术报告题目与摘要

Boundary type GJMS operators

Sun-Yung Alice Chang (Princeton University)

Abstract:We will discuss the connection between the study of non-local operators on Euclidean space to the study of a class of fractional GJMS operators in conformal geometry. The emphasis is on the study of a class of fourth order operators and their third order boundary operators. We will also discuss geometry tools (e.g. met-ric with measures), the positivity property and Sobolev trace inequalities associated with these operators via extension theorems. This is a report of joint works with Jeffrey Case, and separately with Antonio Ache.

 

Some results on positivity in Kahler geometry

Jixiang Fu Fudan University

Abstract: We will talk about the relations between the balanced cone and the Kahler cone of a Kahler manifold, and also talk about Teissier's problem on proportionality of nef and big classes over a compact Kahler manifold. These are joint works with Xiao Jian.

 

Regularity of immersed  hypersurfaces in Riemannian manifolds

Pengfei Guan McGill University

Abstract: We discuss recent joint work with Siyuan Lu on mean curvature estimate for immersed hypersurfaces with nonnegative extrinsic scalar curvature in general ambient space.

Such estimate is related to the Weyl's isometric embedding problem in problem in 3-dimensional Riemannian manifolds. The focus is a degenerate scalar curvature type equation.

The equation is similar to the prescribing curvature equations considered by Caffarelli-Nirenberg-Spruck, except that the prescribed curvature function also depends on the normal of

the surfaces in addition. These type of equations were treated by Guan-Ren-Wang for starshaped surfaces in $R^{n+1}$ and by Spruck-Xiao in space forms.  We will discuss how to establish mean curvature estimate in general Riemannian setting without starshapedness assumption using intrinsic geometry.  We also deal with the degenerate case. If time allows, we will discuss applications to isometric embedding problems and some open problems.

 

Immersed Minimal Surfaces,Orientable or Nonorientable

Robert Gulliver University of Minnesota

Abstract: Minimal surfaces in a Riemannian manifold Mn are surfaces which are stationary for area: the first variation of area vanishes. In this talk we focus on the" false branch point problem" for minimal surfaces ¦M which are of a prescribed topological type, either orientable or nonorientable, mapping Σone-to-one onto a given union G of Jordan curves. We shall show that an area-minimizing surface ¦ factors through an unramified branched immersion ¦ ′:Σ′M via a branched covering :ΣΣ′. In the case n = 3 of codimension one, it follows that ¦ ′ is an immersion on the interior of Σ′.Further, Σ′ will be a surface of lower topological type.

Jesse Douglas in 1939 proved the existence of a minimal surface of a given topological type having smallest area among surfaces of that type,mapping the boundary homeomorphically to a given disjoint union G of Jordan curves, assuming the “Douglas hypothesis” that surfaces of lower topological type with boundary G have strictly larger area. It follows from the result stated above that such a minimizing map will be unrami_ed, and in the case n = 3, an immersion.

In all cases, whether or not the Douglas hypothesis holds, any areaminimizing mapping ¦M 3 defines an immersed surface with smallest area, bounded by the given con_guration of curves, possibly a surface of lower topological type.

 

 

Singular solutions of Yamabe equation

Qing Han University of Notre Dame

Abstract:In a classical paper, Cafferelli, Gidas and Spruck discussed positive solutions of the Yamabe equation, corresponding to the positive scalar curvature of the conformal metrics, with a nonremovable isolated singularity. They proved that solutions are asymptotic to radial singular solutions. Korevaar, Mazzeo, Pacard, and Schoen expanded solutions to the next order. In this talk, we discuss how to expand solutions up to arbitrary order. We also discuss positive solutions of the Yamabe equation, corresponding to the negative scalar curvature of the conformal metrics, that become singular in an (n-1)-dimensional set.

 

A Linear Isoperimetric Inequality for Algebraic and Analytic Varieties

Robert Hardt Rice University

Abstract: In R^n, the classical n dimensional isoperimetric inequality bounds the volume of any smooth region U by c_n [ H^{n-1}(Bdry U)]^n/(n-1) , where c_n is the constant giving equality with U being an n all. In higher codimension, any integral k-1 cycle B in R^n with k < n , is the boundary of some k chain t with mass(t) \leq c_k mass(b)^ k/(k-1) . inside an ambient space x like a complete riemannian manifold, polyhedral complex, or more generally euclidean lipschitz neighborhood retract, every integral boundary bounds some chain satisfying such an isoperimetric inequality with constant c_x depending on x. this relation may fail for singular spaces x with cusps like algebraic varieties.   with t. depauw, we prove a linear isoperimetric inequality valid for cycles of any dimension in a compact set  x  defined by polynomial or real analytic equalities or inequalities. this generalizes earlier work on codimension one boundaries by l.bos -p.milman and b.hua-fh.lin.

 

Regularity Analysis for A Class of Singular Elliptic Equation

Huaiyu Jian Tsinghua University

Abstract: This talk is based on the recent joint works with Xu-Jia Wang at ANU, etc. We prove the optimal regularity and establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.The equations we dealt with include equations from Geometry(Minimal graphs in hyperbolic space, Loewner-Nirenberg problem and hyperbolic affine sphere) as well as from Singualr Stochastic control.

 

Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

 Yanyan Li Rutgers University

Abstract:We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem. This is a joint work with Luc Nguyen.

 

Multiple Green function and Lame equation.

Chang-Shou Lin Taiwan University

 

Abstract How to study the geometry of a flat torus? From the analytical point of view, there seem two simplest ways to begin with. One is the Green function and the other one is the Lam´e equation (a linear second order ODE in complex variable). Surprisingly, these two are deeply related to each other. In this talk, I will report the first page of the story.

 

Superfluid passing an obstacle---Nucleation of Vortices

Fanghua Lin New York University

Abstract:The problem of classical (compressible) fluids passing an obstacle was well-known and studied by many. Roughly speaking, when the velocity of the fluid is small, the flow would be smooth and nothing much would occur(subsonic region). When the fluid velocity is very large, there are shocks and the fluids become rather turbulent and mathematically hard to describe (supersonic). In between, there is a critical speed at which the fluid reach the maximum speed (sound speed for the fluid) at the boundary of the obstacle.

Since a superfluid by definition is frictionless, hence it would not develop shocks. On the other hand, formal arguments imply that the long-wave approximations of superfluid flows (semiclassical limits) would be a classical flow described by the compressible Euler equations. The latter may develop shocks however. A natural question is: how would one explain superfluid flows then?  Reasonings from physics which were also supported by various numerical simulations lead to the so-called vortex nucleations. The aim of this talk is to present a recent joint work with Jun-Cheng Wei on this problem.

 

Neumann Boundary Value Problem for Hessian Equations on Convex  Domain in R^n

Xinan Ma University of Science and Technology of China

Abstract:For the Dirichlet problem on the k- Hessian equation, Caffarelli-Nirenberg-Spruck (1986) obtained the existence of the admissible classical solution when the smooth domain is strictly k-1 convex in R^n. In this talk, we prove the existence of a classical admissible solution to a class of Neumann boundary value problems for k Hessian equations in strictly convex domain in R^n this was asked by Prof.  N. Trudinger in 1987. The methods depends upon the establishment of a priori derivative estimates up to second order.  This is the joint work with Qiu Guohuan.

 

Minimal surfaces in closed and finite volume hyperbolic 3-manifolds.

Harold Rosenberg IMPA, Brazil

Abstract: I will discuss several results.  Laurent Mazet and I proved there exists a closed embedded minimal surface in a finite volume hyperbolic 3-manifold.  When the Heegard genus of the manifold is at least 6, we show the area of the minimal surface is at least 2π.  Ths is also true in closed hyperbolic 3-manifolds.

Laurent Hauswirth, Pascal Collin and I proved that a properly immersed minmal surface of finite topology in a finite volume hyperbolic 3-manifold has total curvature 2π times its Euler characteristic.  We show the annular ends are asymptotic to totally geodesic cusp ends.

I will discuss examples.

 

Metrics of fixed area on high genus surfaces with largest first eigenvalue

Richard Schoen Stanford University and University of California Irvine

Abstract:We will describe the problem of maximizing the first eigenvalue on a closed surface among metrics of a fixed area. This is a nonlocal geometric variational problem whose solutions arise as the induced metrics on certain minimal immersions of the surface into a sphere. We will explain a very general existence theorem and describe questions related to the geometry of the solutions

 

 Self-shrinkers to the mean curvature flow asymptotic to an isoparametric cone

 Joel Spruck Johns Hopkins University

Abstract:  We call a cone C in $R^{n+1}$  an  {\em isoparametric cone} if C is the cone over a compact embedded isoparametric hypersurface in $ \bf{S}^n$. The theory of isoparametic hypersurfaces in $\bf{S}^n$ is extremely rich  and there are  infinitely many distinct classes of examples, each with infinitely many members. In this talk I will construct the unique embedded end of a self-similar shrinking solution of the mean curvature flow  asymptotic to an isoparametric cone C.

 

Global estimates and bubbling analysis of approximate harmonic maps in dimension two.

Changyou Wang Purdue University

Abstract:In this talk, I will discuss the global $W^{2,1}$-estimates for approximate harmonic maps in two dimension, whose tension fields belong to $L\log L$ and the Morrey space $M^{1,a}$ for some $0\le a<2 $, and the bubbling phenomena for weakly convergent sequences of such approximate harmonic maps, including both $l^{2}$ and $l^{2,1}$ identity for their gradients, the $l^1$-identity for their hessians,  and the oscillation identity of the maps.

 

 Geometric Capacity of sets and regularity

Lihe Wang University of Iowa

Abstract: We will study a version of capacity and its rigidity with applications to the regularity of sets.Particularly we will show that if a set has the geometric capacity properties of that of the half space then it has to be the half space.Applications to the regularity of sets are also discussed.

 

Monge-Ampere equations arising in geometric optics

Xu-Jia Wang Australian National University

Abstract:We discuss a Monge-Ampere type equation arising in geometric optics,in the design of a reflection surface such as the reflector antenna.

An interesting phenomenon is that the regularity of the reflection surface depends not only on the light distributions but also on the position of reflector, and the position and curvatures of the object. It was also discovered that the design of reflection surface is a linear optimisation problem in the far field case;and a nonlinear optimisation problem in the near field case.

 

Motion of level sets by general curvature flow

Ling Xiao Rutgers University

Abstract: In this talk, I will first recall some classic results of level set mean curvature flow obtained by Evans and Spruck in the 90s'. Then, I will talk about some related interesting results people have obtained through these years. Finally, I will talk about my results on level set general curvature flow, including an existence theorem and a non-collapsing theorem.

 

Fourth order equations in conformal and CR geometry

Paul Yang Princeton University

Abstract:The 4th-order Q-curvature equation has some progress in dimensions other than four, in  particular the sign of its greens function can be determined under conformally invariant conditions. I will also report on some progress on the Q-prime curvature equation in CR geometry.

 

Special Lagrangian equations

Yu Yuan University of Washington

Abstract:We survey some new and old, positive and negative results on a priori estimates, regularity, and rigidity for special Lagrangian equations with or without certain convexity. The "gradient" graphs of solutions are minimal or maximal Lagrangian submanifolds, respectively in Euclidean or pseudo-Euclidean spaces. In the latter pseudo-Euclidean setting, these equations are just Monge-Ampere equations. Development on the parabolic side (Lagrangian mean curvature flows) will also be mentioned.

 

Properness of F-functional.

Xiaohua Zhu Peking University

Abstract:F-functional  plays an important role in the study of K-stability of Kaehler manifolds.In 1997, Tian gave an  approach to prove Properness of F-functional by using a smooth lemma  from Ricci flow.In this talk, we give another proof of Tian's result  by using the regularity  of complex Monge-Ampere equation. This is a joint work with Tian.

 

 


参会人员名单

姓名

单位

联系方式

Alice S.-Y. Chang

Princeton University

chang@math.princeton.edu

Jixiang Fu

Fudan University

 majxfu@fudan.edu.cn

Pengfei Guan

McGill University

guan@math.mcgill.ca

Robert Gulliver

University of Minnesota

gulliver@umn.edu

Qing Han

University of Notre Dame

qhan@math.pku.edu.cn

Robert Hardt

Rice University

hardt@math.rice.edu

Huaiyu Jian

Tsinghua University

hjian@math.tsinghua.edu.cn

Yanyan Li

Rutgers University

yyli@math.rutgers.edu

Chang-Shou Lin

Taiwan University

cslin@tims.ntu.edu.tw

Fanghua Lin

New York University

linf@cims.nyu.edu

Xinan Ma

USTC

xinan@ustc.edu.cn

Harold Rosenberg

IMPA, Brazil

rosenbergharold@gmail.com

Richard Schoen

Stanford University and University of California Irvine

schoen@math.stanford.edu

Joel Spruck

Johns Hopkins University

Changyou Wang

Purdue University

schoen@math.stanford.edu

Lihe Wang

University of Iowa

Xu-Jia Wang

Australian National University

xu-jia.wang@anu.edu.au

Ling Xiao

Rutgers University

evenling@gmail.com

Paul Yang

Princeton University

yang@math.princeton.edu

Yisong Yang

New York University

yyang@math.poly.edu

Yu Yuan

University of Washington

yuan@math.washington.edu

Xiaohua Zhu

Peking University

xhzhu@math.pku.edu.cn

李工宝

华中师范大学

ligb@mail.ccnu.edu.cn

Guoyi Xu

Tsinghua University

Jiakun Liu

University of Wollongang

Hao Yin

University of Science and Technology of China

Gang Liu

Univertiy of Califorlia, Berkeley

Longzhi Lin

Univertiy of Califorlia, Santa Cruz

Qi Ding

Shanghai Center for Mathematical Sciences, Fudan University

Xiaoli Han

Tsinghua University

Zhou Zhang

University of Sydney

Xuezhang Chen

Nanjing University

Jian Ge

Peking University

Bo Wang

Rutgers University &Beijing Normal University

Lu Wang

University of Wisconsin-Madison

Huichun Zhang

Sun Yat-sen University

Zhenlei Zhang

Capital Normal University

Fang Wang

上海交大

fangwang1984@sjtu.edu.cn

Tang Lang

华中师范大学

lantang@mail.ccnu.edu.cn

Mao-Pei Tsui

maopei@gmail.com

袁伟

中山大学

gnr-x@163.com

刘艳楠

北京工商大学

liuyn@th.btbu.edu.cn

鞠红杰

北京邮电大学

hjju@bupt.edu.cn

纪德生

黑龙江大学

jds4890@163.com

曾令忠

江西师范大学

lingzhongzeng@yeah.net

朱鹏

江苏理工学院

zhupeng@jsut.edu.cn

金亚东

江苏理工学院

方守文

扬州大学

shwfang@163.com

周久儒

扬州大学

zhoujr1982@hotmail.com

郭宇潇

哈尔滨工业大学(威海)

guoyuxiaolove@163.com

牛犇

哈尔滨工业大学(威海)

guoyuxiaolove@163.com

黄勇

湖南大学

huangyong@hnu.edu.cn

徐璐

湖南大学

xulu@hnu.edu.cn

鲁建

浙江工业大学

lujian@zjut.edu.cn

陈传强

浙江工业大学

chuanqiangchen@zjut.edu.cn

侍述军

哈尔滨师范大学

shjshi@163.com

华波波

复旦大学

bobohua@fudan.edu.cn

王培合

曲阜师范大学

peihewang@hotmail.com

矫贺明

哈尔滨工业大学

jiao@hit.edu.cn

李贯锋

哈尔滨工业大学

liguanfeng@hit.edu.cn

王志张

复旦大学

youxiang163wang@163.com

向  妮

湖北大学数学与统计学学院

nixiang@hubu.edu.cn

张德凯

中国科学技术大学

dekzhang@mail.ustc.edu.cn

邓  斌

中国科学技术大学

bingomat@mail.ustc.edu.cn

韦  韡

中国科学技术大学

weisx001@mail.ustc.edu.cn

贾晓含

中国科学技术大学

jia92@mail.ustc.edu.cn

翁良俊

中国科学技术大学

ljweng08@mail.ustc.edu.cn

杨丰瑞

中国科学技术大学

yfrustc@mail.ustc.edu.cn

张永兵

中国科学技术大学

ybzhang@amss.ac.c

林可

重庆大学

shuxuelk@126.com

姚宪忠

重庆大学数学与统计学院

yaoxz416@163.com

Heayong Shin

韩国Chung Ang University

hshin@cau.ac.kr

徐夫义

山东理工大学

fuyixu@163.com

李东升

西安交通大学

lidsh@xjtu.edu.cn

李春和

复旦大学

chli@fudan.edu.cn

康军军

华中科技大学

kjj314@hust.edu.cn

于洋海

华中科技大学

1195454012@qq.com

汤燕斌

华中科技大学

tangyb@hust.edu.cn

吴星

华中科技大学

ny2008wx@163.com

田蓉蓉

华中科技大学

Tianrr06@sina.com

叶运华

嘉应学院数学学院

201301011@jyu.edu.cn 

巫伟亮

嘉应学院数学学院

weilianggood@126.com

Seongtag Kim

Inha University

stkim@inha.ac.kr

Kai Shao

New York University

ks4026@nyu.edu

赵杰

中原工学院

kaifengajie@163.com

张凯

西安交通大学

zkzkzk@stu.xjtu.edu.cn

Kai-Wei Zhao

台湾大学

fmntnge@gmail.com

张伟

兰州大学

zhangw@lzu.edu.cn

董伟松

哈尔滨工业大学

dweeson@gmail.com

崔庆

西南交通大学

qingcui@whu.edu.cn

任长宇

吉林大学

rency@jlu.edu.cn

马飞遥

宁波大学数学系

mafeiyao@nbu.edu.cn

沃维丰

宁波大学数学系

woweifeng@nbu.edu.cn

董荣

西安交通大学

drdxdj689@163.com

冯晓萌

西安交通大学

iamme1208@stu.xjtu.edu.cn

李彩燕

西安交通大学

licaiyan1021@126.com

马姗姗

西安交通大学

mss5221@stu.xjtu.edu.cn

周爽

中南财经政法大学

89619362@qq.com

黃垣熊

台湾大学

d03221001@ntu.edu.tw

Wei-Bo Su

台湾大学

d03221004@ntu.edu.tw

何跃

南京师范大学

heyue@njnu.edu.cn

 张宏

新加坡国立大学

mathongzhang@gmail.com

韩菲

新疆师范大学

137823121@qq.com

王小六

东南大学

xlwang@seu.edu.cn

杨奇林

中山大学

yqil@mail.sysu.edu.cn

王琦

中南财经政法大学

functional@163.com

Huihuang Zhou

上海交大

hhzh33@163.com

廖乃安

重庆大学数学与统计学院

liaonaian@163.com

李栋浩

华中科技大学

Jiehao1021@163.com

李罡

华中科技大学

277085646@qq.com

张华丽

复旦大学

13110180066@fudan.edu.cn

孙玉华

南开大学

sunyuhua@nankai.edu.cn

郭顺滋

四川大学

guoshunzi@yeah.net

艾万君

陈优民

Jianyu Ou

澳门大学

eyes_loki@hotmail.com

冯仁杰

北京大学

renjie@math.pku.edu.cn

厦门大学

bguan@xmu.edu.cn

谭忠

厦门大学

tan85@xmu.edu.cn

邱春晖

厦门大学

chqiu@xmu.edu.cn

张剑文

厦门大学

jwzhang@xmu.edu.cn

厦门大学

chaoxia@xmu.edu.cn