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

Transcription

1 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 9:00-9:45 9:50-10:35 上午 9:00-12:30, 学术报告, 主持人,Yisong Yang Joel Spruck(Johns Hopkins University)Self-shrinkers to the mean curvature flow asymptotic to an isoparametric cone Harold Rosenberg(IMPA, Brazil)Minimal surfaces in closed and finite volume hyperbolic 3-manifolds 10:35-10:55 茶歇 10:55-11:40 11:45-12:30 上午 10:55-12:30, 学术报告, 主持人,Xujia Wang Paul Yang(Princeton University)Fourth order equations in conformal and CR geometry Pengfei Guan(McGill University)Regularity of immersed hypersurfaces in Riemannian manifolds 12:30-14:30 午餐 休息 14:30-15:15 15:20-16:05 下午 14:30-16:05, 学术报告, 主持人,Yanyan Li Alice S.-Y. Chang(Princeton University)Boundary type GJMS operators Jixiang Fu(Fudan University) Some results on positivity in Kahler geometry 16:05-16:25 茶歇 16:25-17:10 下午 16:25-18:00, 学术报告, 主持人,Gongbao Li 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

2 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 9:30-10:15 上午 9:30-10:15, 学术报告, 主持人,Pengfei Guan Chang-Shou Lin(Taiwan University)Multiple Green function and Lame equation 10:15-10:35 茶歇 10:35-11:20 11:25-12:10 上午 10:35-12:10, 学术报告, 主持人,Changshou Lin Lihe Wang(University of Iowa)Geometric capacity of sets and regularity Robert Gulliver(University of Minnesota) Immersed minimal surfaces,orientable or nonorientable 12:10-14:30 午餐 休息 14:30-15:15 15:20-16:05 下午 14:30-18:00, 学术报告, 主持人,Paul Yang Fanghua Lin(New York University)Superfluid passing an obstacle---nucleation of vortices 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

3 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 9:00-9:45 9:50-10:35 上午 9:00-10:35, 学术报告, 主持人,Robert Hardt Richard Schoen(Stanford University and University of California Irvine)Metrics of fixed area on high genus surfaces with largest first eigenvalue Ling Xiao(Rutgers University)Motion of level sets by general curvature flow 10:35-10:55 茶歇 10:55-11:40 11:45-12:30 上午 10:55-12:30, 学术报告, 主持人,Lihe Wang Xujia Wang(Australian National University)Monge-Ampere equations arising in geometric optics 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 15:20-16:05 Robert Hardt(Rice University)A linear isoperimetric inequality for algebraic and analytic varieties Qing Han(University of Notre Dame) Singular solutions of Yamabe equation

4 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 M n 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 ƒ:σ M which are of a prescribed topological type, either

5 orientable or nonorientable, mapping Σone-to-one onto a given union Γ of Jordan curves. We shall show that an area-minimizing surface ƒ factors through an unramified branched immersion ƒ :Σ M via a branched covering :Σ Σ. In the case n = 3 of codimension one, it follows that ƒ 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 Γ of Jordan curves, assuming the Douglas hypothesis that surfaces of lower topological type with boundary Γ 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 ƒ:σ 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.

6 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

7 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 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

8 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)

9 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.

10 参会人员名单 姓名单位联系方式 Alice S.-Y. Chang Princeton University Jixiang Fu Fudan University Pengfei Guan McGill University Robert Gulliver University of Minnesota Qing Han University of Notre Dame Robert Hardt Rice University Huaiyu Jian Tsinghua University Yanyan Li Rutgers University Chang-Shou Lin Taiwan University Fanghua Lin New York University Xinan Ma USTC Harold Rosenberg IMPA, Brazil Richard Schoen Joel Spruck Stanford University and University of California Irvine Johns Hopkins University Changyou Wang Purdue University Lihe Wang University of Iowa Xu-Jia Wang Australian National University Ling Xiao Rutgers University Paul Yang Princeton University Yisong Yang New York University Yu Yuan University of Washington Xiaohua Zhu Peking University 李工宝华中师范大学 Guoyi Xu Jiakun Liu Hao Yin Gang Liu Longzhi Lin Tsinghua University University of Wollongang University of Science and Technology of China Univertiy of Califorlia, Berkeley Univertiy of Califorlia, Santa Cruz

11 Qi Ding Xiaoli Han Zhou Zhang Xuezhang Chen Jian Ge Bo Wang Lu Wang Huichun Zhang Zhenlei Zhang Shanghai Center for Mathematical Sciences, Fudan University Tsinghua University University of Sydney Nanjing University Peking University Rutgers University &Beijing Normal University University of Wisconsin-Madison Sun Yat-sen University Capital Normal University Fang Wang 上海交大 Tang Lang 华中师范大学 Mao-Pei Tsui 袁伟中山大学 刘艳楠北京工商大学 鞠红杰北京邮电大学 纪德生黑龙江大学 曾令忠江西师范大学 朱鹏江苏理工学院 金亚东 江苏理工学院 方守文扬州大学 周久儒扬州大学 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

12 王志张 复旦大学 向妮 湖北大学数学与统计学学院 张德凯 中国科学技术大学 邓斌 中国科学技术大学 韦韡 中国科学技术大学 贾晓含 中国科学技术大学 翁良俊 中国科学技术大学 杨丰瑞 中国科学技术大学 张永兵 中国科学技术大学 林可 重庆大学 姚宪忠 重庆大学数学与统计学院 Heayong Shin 韩国 Chung Ang University 徐夫义 山东理工大学 李东升 西安交通大学 李春和 复旦大学 康军军 华中科技大学 于洋海 华中科技大学 汤燕斌 华中科技大学 吴星 华中科技大学 田蓉蓉 华中科技大学 叶运华 嘉应学院数学学院 巫伟亮 嘉应学院数学学院 Seongtag Kim Inha University Kai Shao New York University 赵杰 中原工学院 张凯 西安交通大学 Kai-Wei Zhao 台湾大学 张伟 兰州大学 董伟松 哈尔滨工业大学 崔庆 西南交通大学 任长宇 吉林大学 马飞遥 宁波大学数学系 沃维丰 宁波大学数学系 董荣 西安交通大学 冯晓萌 西安交通大学

13 李彩燕 西安交通大学 马姗姗 西安交通大学 周爽 中南财经政法大学 黃垣熊 台湾大学 Wei-Bo Su 台湾大学 何跃 南京师范大学 张宏 新加坡国立大学 韩菲 新疆师范大学 王小六 东南大学 杨奇林 中山大学 王琦 中南财经政法大学 Huihuang Zhou 上海交大 廖乃安 重庆大学数学与统计学院 李栋浩 华中科技大学 李罡 华中科技大学 张华丽 复旦大学 孙玉华 南开大学 郭顺滋 四川大学 艾万君陈优民 Jianyu Ou 澳门大学 冯仁杰 北京大学 关波 厦门大学 谭忠 厦门大学 邱春晖 厦门大学 张剑文 厦门大学 夏超 厦门大学