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“椭圆型偏微分方程及其应用”
学术研讨会
会议手册
上海交通大学
2019 年 11月 15 日-11 月 17日
目 录
组织委员会 ....................................................................................... 1
会议报到及安排 ............................................................................... 1
会议注册报到............................................................................................. 1
住宿安排 ..................................................................................................... 1
餐饮安排 ..................................................................................................... 1
酒店-学院班车 ........................................................................................... 2
无线网络 ..................................................................................................... 2
学院位置 ..................................................................................................... 2
会议日程 ........................................................................................... 4
报告摘要 ........................................................................................... 7
参会人员 ......................................................................................... 14
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
1
组织委员会
1 李从明（主席） 上海交通大学
2 王维克 上海交通大学
3 邓引斌 华中师范大学
4 郭玉劲 华中师范大学
5 来米加（联系人） 上海交通大学
6 王芳（联系人） 上海交通大学
会议报到及安排
会议注册报到
会议时间：2019年 11月 15日-17日
报到时间：2019 年 11 月 15 日 8:30-9:00
会议地点：上海交通大学理科楼 6 号楼 706 室
上海交通大学闵行校区地址：上海市闵行区东川路 800号
住宿安排
酒店：曼哈顿酒店（闵行店）
地址：上海市闵行区鹤庆路 900 号 5-6 号楼（近碧江路）
电话：021- 67281666
我们已为您预订好房间（含早餐），请使用身份证件登记入住。早餐供应时
间为上午 7:00 – 9:30。
餐饮安排
会议提供11月15日、16日和17日午餐（上海交通大学学术活动中心中餐厅）
以及11月14日（曼哈顿酒店湘遇食年餐厅）、15日（丰收日碧江广场店）、
16日和17日（上海交通大学留园餐厅）晚餐
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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酒店-学院班车
11月15日、16日和17日会有班车往返酒店和学院，具体安排如下：
日期 时间 路线
11 月 15 日 8:30 曼哈顿酒店 – 数学科学学院
18:00 数学科学学院 – 餐厅
11 月 16 日和
17 日
8:00 曼哈顿酒店 – 数学科学学院
19:30 （以实际结束时
间为准）
餐厅 – 曼哈顿酒店
无线网络
无线网名：LargeConference
密码：12345678
学院位置
数学科学学院（见下一页地图）
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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会议日程
11 月 15 日，周五
08:30 - 09:20 会议注册
09:20 - 09:30 开幕式
上午 Session I 主持：邓引斌（华中师范大学）
09:30 - 10:20 汪徐家（澳大利亚国立大学） 题目：Optimal transport and related topics
10:30 -11:00 茶歇
上午 Session II 主持：王维克（上海交通大学）
11:00 -11:50 潘兴斌（华东师范大学） 题目：Div-Curl System with Potential and Maxwell-Stokes System with Natural Boundary Condition
12:00 -14:30 午餐（上海交通大学学术活动中心中餐厅）
下午 Session I 主持：桂长峰（德克萨斯大学圣安东尼奥分校）
14:30 -15:20 杨孝平（南京大学） 题目：Some Properties of Solutions to Biased Infinity Laplacian Equations
15:30 -16:00 茶歇
下午 Session II 主持：周风（华东师范大学）
16:00 -16:50 刘兆理（首都师范大学） 题目：Improvements and generalizations of Clark's
theorem and applications
17:00 -17:50
郭玉劲（华中师范大学） 题目： On the Nonexistence of Vortex States for Rotating Bose-Einstein Condensates with Attractive Interactions
18:00 晚餐（丰收日碧江广场店）
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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11 月 16 日，周六
上午 Session I 主持：张立群（中国科学院数学与系统科学研究院）
08:30 -09:20 丁彦恒（中国科学院） 题目：变分法与交叉科学
09:30 -10:00 茶歇与集体照
上午 Session II 主持：朱熹平（中山大学）
10:00 -10:50 彭双阶（华中师范大学） 题目：The number of positive solutions to the Brezis-Nirenberg problem
11:00 -11:50 李东升（西安交通大学） 题目：Existence and Boundary Asymptotic Behavior of Hessian Equations
12:00 -14:30 午餐（上海交通大学学术活动中心中餐厅）
下午 Session I 主持：顾永耕（湖南师范大学）
14:30 -15:20 桂长峰（德克萨斯大学圣安东尼奥分校） 题 目 ： New Developments in Sphere Covering Inequality
15:30 -16:00 茶歇
下午 Session II 主持：刘兆理（首都师范大学）
16:00 -16:50 周焕松（武汉理工大学） 题目：一类带强制位势的非局部椭圆型特征值问题
17:00 -17:50 郭宗明（河南师范大学） 题目：Isolated singularities of some weighted second
and fourth order elliptic equations
18:00 晚餐（上海交通大学留园餐厅）
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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11 月 17 日，周日
上午 Session I 主持：朱长江（华南理工大学）
08:30 -09:20 曹道民（中国科学院数学与系统科学研究院） 题目：不可压欧拉方程的定常涡环解
09:30 -10:00 茶歇
上午 Session II 主持：汪徐家（澳大利亚国立大学）
10:00 -10:50 杨健夫（江西师范大学） 题目：Normalized solutions and mass concentration for supercritical nonlinear Schr\"{o}dinger equations
11:00 -11:50 郭玉霞（清华大学） 题目： Nondegeneracy for multiple solutions and applications
12:00 -14:30 午餐（上海交通大学学术活动中心中餐厅）
下午 Session I 主持：郭玉劲（华中师范大学）
14:30 -15:20 严树森（华中师范大学） 题目：Excited states on Bose-Einstein condensates with attractive interactions
15:30 -16:00 茶歇
下午 Session II 主持：李从明（上海交通大学）
16:00 -16:50 戴求亿（湖南师范大学） 题目： Iterative method for Kirchhoff-Carrier type
equations and its applications
17:00 -17:50 自由讨论
18:00 晚餐（上海交通大学留园餐厅）
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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报告摘要
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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不可压欧拉方程的定常涡环解
曹道民
中国科学院数学与系统科学研究院
报告人将报告近年来在不可压欧拉方程定常涡解方面的研究，特别地要介绍在二
维时涡补丁解（vortex patch)和 三维时涡环解（vortex ring）存在性和唯一性方
面的结果。报告人将讲述涡补丁解的存在唯一性与 Kirchhoff - Routh 函数临界
点之间的联系，而 Kirchhoff - Routh 临界点又与方程所在区域的几何性质密切
相关。 报告人主要介绍的结果是来源于和郭玉霞、彭双阶、严树森合作的论文
及和王国栋、詹伟城合作的论文。
Iterative method for Kirchhoff-Carrier type equations and its
applications
戴求亿
湖南师范大学
Let $A(s, t)$ be a continuous function with a positive lower bound $m$, and
$\Omega$ be a bounded domain in $R^N$. In this short note, we propose an
iterative procedure for finding nonnegative solutions of the following Kirchhoff-
Carrier type equations
\begin{equation*}\left\{
\begin{array}{ll}
-A(\|u\|_p, \|\nabla u\|_2)\Delta u=g(x, u) & x\in\Omega,\\
u=0 & x\in\partial\Omega.
\end{array}\right.
\end{equation*}
The main advantage of our procedure is that the convergent proof of the
iterative sequence depends only on comparison principle of the Laplace
operator instead of comparison principle of Kirchhoff-Carrier type operator itself.
Therefore, we almost need no restrictions on $A(s, t)$ except for continuous
and a positive lower bound. This removes away the monotonicity assumption
of $A(s, t)$ used in most papers based on sub-supersolution method. As
applications of the abstract result obtained by our iterative method, some
concrete examples are also studied.
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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变分法与交叉科学
丁彦恒
中国科学院
一般半线性问题的变分框架；特别如 Hamilton 系统、反应扩散系统、Dirac 方
程等；进而谈一些变分方法对交叉科学的作用。
New Developments in Sphere Covering Inequality
桂长峰
德克萨斯大学圣安东尼奥分校
In this talk, I will review the Sphere Covering Inequality and present some new
forms of the inequality, including a version with singular terms on more general
surfaces as well as a dual version. I will also show their applications in the study
of singular Liouville-type problems with super harmonic weights. In particular,
new uniqueness results for solutions of the singular mean field equation both
on spheres and on bounded domains will be discussed. Some new symmetry
results for the spherical Onsager vortex equation will also be presented.
On the Nonexistence of Vortex States for Rotating Bose-Einstein
Condensates with Attractive Interactions
郭玉劲
华中师范大学
In this talk, we focus on ground states of two-dimensional attractive Bose-
Einstein condensates (BEC) in a rotating trap, which can be described by the
complex-valued Gross-Pitaevskii energy functional. We discuss the
classification of the existence and nonexistence for ground states, which
depends on the trap’s rotational velocity V and the attractive strengthen N of
cold atoms as well. As N approaches to a critical value and the rotational
velocity V is low, we prove that, up to the phase rotation, all ground states are
real-valued,unique and free of vortices. This gives the nonexistence of vortex
states for rotating BEC.
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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Nondegeneracy for multiple solutions and applications
郭玉霞
清华大学
We consider the following prescribed scalar curvature equations in $\R^N$
\begin{equation}\label{eq}
- \Delta u =K(|y|)u^{2^*-1},\quad u>0 \ \ \mbox{in} \ \R^N, \ \ \
u \in D^{1, 2}(\R^N),
\end{equation}
where $K(r)$ is a positive function, $2^*=\frac{2N}{N-2}$. We will talk about a
non-degeneracy result for the positive multi-bubbling solutions by using the
local Pohozaev identities. Then we use this non-degeneracy result to glue
together bubbles with different concentration rate to obtain new solutions. This
is joint work with Musso, Peng and Yan.
Isolated singularities of some weighted second and fourth order elliptic
equations
郭宗明
河南师范大学
We consider asymptotic behavior at the isolated singularities of positive solutions of some
weighted second and fourth order elliptic equations with subcritical or critical exponent.
Our results can be used to study the behavior at the singularities of positive solutions of
some equations on singular manifolds with conical metrics.
Existence and Boundary Asymptotic Behavior of Hessian Equations
李东升
西安交通大学
In this talk, we will establish the existence of viscosity solutions of Hessian equations with
singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The
asymptotic results generalize those for Poisson equations and are more precise than
obtained from Hopf lemma. We will also establish the existence of large solutions of
Hessian equations and obtain a new boundary asymptotic behavior of solutions.
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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Improvements and generalizations of Clark's theorem and applications
刘兆理
首都师范大学
In critical point theory, Clark's theorem states as follows. Let $X$ be a Banach space,
$\Phi\in C^1(X,\mathbb R)$. Assume $\Phi$ satisfies the (PS) condition, is even and
bounded from below, and $\Phi(0) = 0$. If for any $k\in\mathbb N$, there exists a $k$-
dimensional subspace $X^k$ of $X$ and $\rho_k>0$ such that $\sup_{X^k\cap
S_{\rho_k}}\Phi<0$, where $S_\rho=\{u\in X|\ \|u\|=\rho\}$, then $\Phi$ has a sequence of
critical values $c_k<0$ satisfying $c_k\to 0$ as $k\to \infty$. We improve Clark's theorem
by showing that under the assumptions of Clark's theorem $\Phi$ has a sequence of critical
points $u_k$ such that $\Phi(u_k)\leq0$ and $u_k\to 0$ as $k\to \infty$. We also generalize
Clark's theorem by replacing the $C^1$ smoothness, the boundedness from below, and
the (PS) condition with weaker assumptions respectively. The new results produce infinitely
many solutions to various nonlinear equations under quite general conditions. This is joint
work with Shaowei Chen and Zhi-Qiang Wang.
Div-Curl System with Potential and Maxwell-Stokes System with Natural
Boundary Condition
潘兴斌
华东师范大学
In this talk we examine the div-curl system with potential, and the boundary value problem
of the Maxwell-Stokes system under natural boundary condition. One of the main features
of these systems is that the equations contain an unknown potential term, which represents
the effect of domain topology. Solvability of these systems depends not only on the
structure of the equations and the type of the boundary conditions, but also on the domain
topology. We show existence of solutions of these systems by using variational methods
together with the modified de Rham lemma, and by the method of compact operators.
Regularity of the solutions and the eigenvalue problem are also discussed.
The number of positive solutions to the Brezis-Nirenberg problem
彭双阶
华中师范大学
In this talk we are concerned with the well-known Brezis-Nirenberg problem. By local
Pohozaev identities and blow-up analysis, we obtain the structure and local uniqueness of
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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blow-up solutions to the Brezis-Nirenberg problem. From these and the analysis of the
blow-up solutions, we give a description of the number of positive solutions to the Brezis-
Nirenberg problem for small positive $\varepsilon$, which depends on the Green's function
of the domain $\Omega$.
Optimal transport and related topics
汪徐家
澳大利亚国立大学
Optimal transport has found applications in many different areas, in particular it is a useful
tool in image process and machine learning. Optimal transport is closely related to the
Monge-Ampere equation. Its potential function satisfies a second boundary condition of
the Monge-Ampere equation. By studying the problem one can prove the existence,
uniqueness, and regularity of optimal mappings. In this talk, we will explain application of
optimal transport in machine learning, and review some recent results on the regularity in
optimal transportation, in particular those obtained by my collaborators and myself.
Normalized solutions and mass concentration for supercritical nonlinear
Schr\"{o}dinger equations
杨健夫
江西师范大学
In this talk, we deal with the existence and concentration of normalized
solutions to the supercritical nonlinear Schr\"{o}dinger equation
\begin{equation}\label{eq:0.1} \left\{ \begin{array}{l}
-\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad \mathbb{R}^2,\\
\int_{\mathbb{R}^2}|u|^2\,dx =1,\\
\end{array}\right. \end{equation}
where $\mu_q$ is the Lagrange multiplier. We show that for $q>2$ close to $2$,
problem\eqref{eq:0.1} admits two solutions: one is the local minimal solution
$u_q$ and another one is the mountain pass solution $v_q$. Furthermore, we
study the limiting behavior of $u_q$ and $v_q$ when $q\to 2_+$. Particularly,
we describe precisely the blow-up formation of the excited state $v_q$.
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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Some Properties of Solutions to Biased Infinity Laplacian Equations
杨孝平
南京大学
In this talk, we will discuss a kind of infinity Laplacian equations arising from the biased
tug-of-war in random games. Various properties of the solutions for such equations,
including the gradient estimates, the Harnack inequalities for both nonnegative $u$ and
$|Du|$, the principal eigenvalue are established. We will also prove some existence and
uniqueness results. This is some joint work with Liu Fang.
Excited states on Bose-Einstein condensates with attractive interactions
严树森
华中师范大学
We study the Bose-Einstein condensates (BEC) in two or three dimensions with attractive
interactions, described by $L^{2}$ constraint Gross-Pitaevskii energy functional. First, we
give the precise description of the chemical potential of the condensate $\mu$ and the
attractive interaction $a$. Next, for a class of degenerated trapping potential with non-
isolated critical points, we obtain the existence and the local uniqueness of the excited
states by accurately analyzing the location of the concentrated points and the
Lagrange multiplier. To our best knowledge, not any result was obtained for the excited
states of BEC in mathematics. Also, our results on degenerated trapping potential with
non-isolated critical points are also new even for the ground states of BEC. This is a joint
work with Peng Luo and Shuangjie Peng.
一类带强制位势的非局部椭圆型特征值问题
周焕松
武汉理工大学
本报告将主要介绍报告人及其合作者关于一类带强制位势的 Kirchhoff 型椭圆型方程特征值
问题的有关结果，如解的存在性、解的渐近性态等。
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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参会人员
1 曹道民 中国科学院数学与系统科学研究院
2 戴求亿 湖南师范大学
3 邓师瑾 上海交通大学
4 邓引斌 华中师范大学
5 丁彦恒 中国科学院
6 方北香 上海交通大学
7 顾永耕 湖南师范大学
8 桂长峰 德克萨斯大学圣安东尼奥分校
9 郭玉劲 华中师范大学
10 郭玉霞 清华大学
11 郭宗明 河南师范大学
12 何其涵 广西大学
13 金石 上海交通大学
14 来米加 上海交通大学
15 李从明 上海交通大学
16 李东升 西安交通大学
17 李亚纯 上海交通大学
18 刘成杰 上海交通大学
19 刘兆理 首都师范大学
20 潘兴斌 华东师范大学
21 彭双阶 华中师范大学
22 皮慧荣 广西大学
23 施小丁 北京化工大学
24 汪徐家 澳大利亚国立大学
25 王芳 上海交通大学
“椭圆型偏微分方程及其应用”学术研讨会 2019.11.15-11.17
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26 王海涛 上海交通大学
27 王维克 上海交通大学
28 王晓东 上海交通大学
29 王亚光 上海交通大学
30 武乐云 上海交通大学
31 谢春景 上海交通大学
32 谢峰 上海交通大学
33 徐永忠 上海交通大学
34 许德良 上海交通大学
35 严树森 华中师范大学
36 杨健夫 江西师范大学
37 杨孝平 南京大学
38 杨雄锋 上海交通大学
39 杨义虎 上海交通大学
40 叶东 华东师范大学
41 张立群 中国科学院数学与系统科学研究院
42 赵会江 武汉大学
43 周春琴 上海交通大学
44 周风 华东师范大学
45 周焕松 武汉理工大学
46 朱苗苗 上海交通大学
47 朱熹平 中山大学
48 朱长江 华南理工大学
