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

6月16日，海韵园数学物理大楼117

6月17日，海韵园数学物理大楼117

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 ¦:Σ→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 :Σ→Σ′. 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 ¦: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.

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