Resume

A new Journal

椭圆与抛物方程杂志

JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS

投稿：

http://www.orthogonal-editions.com/JEPE.html

部分学术活动备忘

2013年研究生短课程与学术报告

Mini-Courses for Graduate Students and Seminar Lectures

Mathematical Analysis of Nematic Liquid Crystals in Landau-de Gennes Theory

Prof. Jinhae Park

（Chungnam National University， South Korea）

2013年10月11日（星期五）下午1：30-2：30

华东师大闵行校区数学楼102报告厅

摘要：In this talk, we give a brief introduction to Ferroelectric Liquid Crystals and Landau-de Gennes Energy for Liquid Crystals which generalizes the classical Oseen-Frank energy. After that, we discuss singularities of degree 1/2 in Nematic Liquid Crystals.

NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai

Special Seminar Series

Title: Literature and Working Seminars on Eigenvalues and Eigenfunctions

Organizers: Prof. Fanghua Lin and Prof. Xingbin Pan

Time: Every Thursday (Commencing from September 12), 2:30pm - 4:30pm, Fall Semester 2013

Venue: Room 371, Geography Building, 3663 North Zhongshan Road, (华东师范大学中山北路校区，地理楼371室)

Description: The NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai will host literature & working seminars on various research topics for students and post-doctoral researchers. Faculty and visitors are welcome to participate. We would also like to invite all those with expertise on the topics of discussion to give survey lectures. The seminar series for this semester will focus on “Eigenvalues and Eigenfunctions for Elliptic Operators”. Professor Fanghua Lin will be the main speaker of the first seminar which will be held on Thursday, September 12th.

偏微分方程短课程

Theory of Elliptic, Periodic Homogenization

林芳华教授 (Courant Institute, NYU）

6月17日下午 2:00-4:30，

6月18日上午 9:00-11:30，

6月19日下午 2:00-4:30，

6月20日上午 9:00-11:30，

6月25日上午9:00-11:30,

6月26日下午2:00-4:30，

6月27日上午9:00-11:30，

6月28日下午2:00-4:30，

地点：华东师大中山北路校区理科大楼A座1414，或A座1510 (17日、19日在A座 510报告厅)

闵行校区前往中北校区听课交通安排将另行通知。

摘要： After a brief introduction, we shall describe first a few important examples of homogenization: an ODE case and the WKB formal expansions for the periodic case. Then we discuss some general theory of abstract H-(and G-) convergences. A more detailed exposition would follow on periodic homogenizations (including uniform estimates and applications, optimal convergence rates and applications, fundamental solutions and boundary layers...). Then we continue with homogenizations of Hamilton-Jacobi, fully nonlinear equations, and free interfaces ... random homogenizations and some open problems. Further topics on the theory of H-measures (micro-local measures) and compensate compactness and convex integrations may be explored.

Remarks: Lectures will be mostly self-contained. Students with some basic knowledge of real variables, elliptic PDEs, Hilbert spaces and Fourier integrals shall be able to follow the lectures.

Random Unitary Matrices with Structure

Prof. Brendan Farrell

California Institute of Technology, USA

Farrell@cms.caltech.edu

2013年5月30日（星期四）下午3：10-4：10

华东师大闵行校区数学楼102报告厅

摘要：Matrices with uniform distribution on the set of unitary matrices are the cannonical type of matrix to work with in many settings. For example, selecting a random subset of the columns of such a matrix gives a uniform distribution on subspaces of a certain dimension. Yet, there are distributions on the set of unitary matrices that allow for more structure and less randomness while still retaining important properties of uniformly distributed random unitary matrices. We present recent results for unitary matrices constructed using a Hadamard matrix and a signed permutation matrix or Kronecker products of a uniformly distributed unitary matrix. The structured approach also very naturally leads to formulations of discrete uncertainty principles. This is joint work with G. Anderson and R.R. Nadakuditi.

Mathematical modeling of diffusion in heterogeneous environment

Professor Yong Jung Kim

Korea Advanced Institue of Science and Technology

2013年2月28日（星期四）下午3：10-4：10

华东师大闵行校区数学楼102报告厅

摘要：It was Einstein who provided the theory to prove the existence of atoms and the theory is about diffusion phenomenon. In this talk his theory is introduced from a mathematical modeling view point and then extended to explain the thermal diffusion phenomenon.

2012年研究生短课程与学术报告

Mini-Courses for Graduate Students and Seminar Lectures

研究生短课程

Mean Curvature Motion: Homogenization and Traveling Waves

Professor Regis Monneau

École des Ponts Paris Tech, France

(http://cermics.enpc.fr/~monneau)

时间： 2012年7月5、6、9、11、13、14日，下午1：30-4：00

地点：华东师大闵行校区数学楼102报告厅

摘要： In these lectures, we are interested in the dynamics of curves or surfaces moving with Mean Curvature Motion (MCM). We will present recent results about both Homogenization of forced MCM and the construction of traveling waves.

日程：

7月5日（周四）: Introduction to Mean Curvature Motion (MCM)
7月6日（周五）: Introduction to Homogenization (I)
7月9日（周一）: Introduction to Homogenization (II)
7月11日（周三）: Homogenization of MCM
7月13日（周五）: Traveling waves for MCM (I)
7月14日（周六）: Traveling waves for MCM (II)

参考文献：

[1] Y. Giga, Surface evolution equations. A level set approach, Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, (2006).

[2] P.-L. Lions, P.E. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 5, 667-677.

[3] R. Monneau, J.-M. Roquejoffre, V. Roussier-Michon, Travelling graphs for the forced mean curvature motion in an arbitrary space dimension, http://cermics.enpc.fr/~monneau/MRR-220212.pdf

Uniqueness of Heat Flow of Harmonic Maps in Higher Dimensions

Professor Changyou Wang

University of Kentucky

2012年6月12日（星期二）下午4：15-5：15

华东师大闵行校区数学楼102报告厅

摘要：. In this talk, we will discuss the uniqueness result for the initial-boundary value problem of heat flow of harmonic maps that enjoy small renormalized energies. Both convexity and unique limit at time infinity for such a class of heat flow of harmonic maps are also obtained. As a byproduct, we obtain the uniqueness of heat flow of harmonic maps in the Serrin's class. Some application to the hydrodynamic flow of nematic liquid crystals will also be discussed.

Smoothness and Long-time Behavior of the Spreading Front Determined

by a Nonlinear Free Boundary Problem

Professor Yihong Du

University of New England, Australia

2012年5月15日（星期二）下午3：30-4：30

华东师大闵行校区数学楼102报告厅

摘要： We consider the spreading of species governed by a free boundary model. In one space dimension and in the radially symmetric case with a logistic nonlinear term, it is known that this model exhibits a spreading-vanishing dichotomy. In this talk we discuss the non-radially symmetric case with more general nonlinearities. By establishing suitable regularity of the free boundary, we show that the spreading-vanishing dichotomy still holds. Moreover, when spreading happens, the normalized free boundary approaches the unit sphere as time goes to infinity. For logistic nonlinearity we obtain a complete extension of the previous results in the radially symmetric case to the general case. This is joint work with Hiroshi Matano and Kelei Wang.

2011年研究生短课程与学术报告

Mini-Courses for Graduate Students and Seminar Lectures

Workshop on Nonlinear Partial Differential Equations

--China-Japan Joint Project for Young Mathematician

TIME: November 3-5, 2011.

PLACE: Department of Mathematics and Center for PDE, East China Normal University,

SPEAKERS:

Hayato Chiba (Kyushu University, chiba@imi.kyushu-u.ac.jp)

Tetsuya Ishiwata (Shibaura Institute of Technology, tisiwata@shibaura-it.ac.jp )

Tatsuki Kawakami (Osaka Prefecture University, kawakami@ms.osakafu-u.ac.jp)

Yuxiang Li ( Southeast University, lieyx@seu.edu.cn, lieyuxiang@yahoo.com.cn)

Zhaoli Liu (Capital Normal University, zliu@mail.cnu.edu.cn)

Bendong Lou (Tongji University, blou@mail.tongji.edu.cn)

Xinan Ma (Chinese University of Science and Technology, xinan@ustc.edu.cn)

Hideki Murakawa (Kyushu University, murakwa@math.kyushu-u.ac.jp)

Takashi Okuda (Kwansei Gakuin University, tokuda@mc-jma.go.jp )

Masahiko Shimojo (Meiji University, shimojotw@gmail.com )

Kanako Suzuki (Tohoku University, kasuzu@m.tohoku.ac.jp )

Zhitao Zhang (Chinese Academy of Sciences, zzt@math.ac.cn)

Jinyu Li (Postdoc, Centerfor PDE, East China Normal University)

Dan Liu (Postdoc, Centerfor PDE, East China Normal University)

Liping Wang (East China Normal Hniversity, lpwang@math.ecnu.edu.cn)

Xingfei Xiang (Ph D. student, East China Normal University)

Yanyan Zhang (East China Normal Hniversity, yyzhang@math.ecnu.edu.cn)

Chunyi Zhao (East China Normal University, cyzhao@math.ecnu.edu.cn)

ORGANIZERS: Y. Morita (Ryukoku Univ.), W.-M. Ni (ECNU and Minnesota), X. Pan (ECNU) and F. Zhou (ECNU)

SECRETARY: Hongyan Zhang, ECNU,

Lectures on Oblique Derivative Problems for Elliptic and Parabolic Differential Equations

(椭圆与抛物方程斜微商问题短课程)

Prof. Gary Lieberman

(Iowa State University)

http://www.public.iastate.edu/~lieb/

Date: July. 20 (Wed.) July. 22 (Fri.)， July. 25 (Mon.)

July. 27 (Wed.) ， July. 29 (Fri.)， Aug. 1 (Mon.)

Time: 1:30--4:00 P.M.

Vanue: 华东师大闵行校区数学系102报告厅

Finite element methods for Ginzburg-Landau equations of superconductivity

Professor Lin Ping

University of Dundee, UK

2011年6月23日（星期四）下午1：00-2：00

华东师大闵行校区数学楼102报告厅

摘要： We use continuous finite element method to show the boundary layer behavior of the order parameter of the minimizing solutions of the Ginzburg-Landau equations. This result confirms the mathematical analysis on boundary effect to the concentration behavior of surface superconductivity as well as the distribution of the vortices.

Poiseuille flow and apparent viscosity of nematic liquid-crystals

Professor Liu Weishi

University of Kansas

2011年6月16日（星期四）下午3:20—4:20

华东师大闵行校区数学系102报告厅

摘要：In this talk, we will study the steady Poiseuille flow of nematic liquid-crystals. A good understanding of this flow can be obtained due to a rather surprising observation that takes advantage of a special structure of this specific problem. As an application of the dynamics of the Poiseuille flow, we provide an approximation for the apparent viscosity -- an important characteristic quantity for fluids.

Asymptotics of the interface of a Hele-Shaw flow with multiple point sources

Dr. Michiaki Onodera

National Taiwan University

2011年3月15日（星期二）下午1:00—2:00

华东师大闵行校区数学系102报告厅

摘要：We study the asymptotic behavior of the interface of a Hele-Shaw flow produced by the injection of fluid from finitely many points at different speeds. We prove that, as time goes to infinity, the interface approaches an expanding circle centered at the barycenter of the injection points with weights proportional to the injection rates. The distances from the barycenter to the boundary points are estimated from above and below. The proof is based on a precise estimate of quadrature domains of the measure in the case where a, b>0 and a+b is sufficiently large. Here denote the Dirac measures at i, -i respectively. The definition and basic ideas of quadrature domains will be provided in the talk. Combining an induction argument with the estimate yields the result

2010年研究生短课程与学术报告

Mini-Courses for Graduate Students and Seminar Lectures

Global Minimizers for a p-Ginzburg-Landau Energy

演讲人：Professor Yaniv Almog

（Louisiana State University）

2010年12月16日（星期四）下午3：00-4：00

华东师大闵行校区数学楼102报告厅

摘要： We consider here minimization of the p-Ginzburg-Landau energy functional for p>2 in the space of w^{1,p}_{loc} functions whose degree along circles of large radii is 1. We first review previous works where we prove existence of minimizers and obtain some of the properties of radially symmetric minimizers. We then report some recent results showing that as p goes to infinity the global minimizer is at least close in some L^\infty sense to a radially symmetric minimizer. This is a Joint work with L. Berlyand, D. Golovaty and I.Shafrir.

“I Want to Be a Scientist!”

演讲人： Professor Chong Chi Tai

(National University of Singapore）

2010年12月13日（星期一）下午1：30-2：30

华东师大闵行校区数学楼102报告厅

摘要: This is a talk for a general audience interested in science. It will discuss several key elements that define a scientist, citing historical examples from the past and present.

Interactive Mesh and Image Cutting

Using Constrained Random Walks

演讲人： Professor Zheng Jianmin

(Nanyang University of Technology, Singapore)

2010年12月10日（星期五）下午2：30-3：30

华东师大闵行校区数学楼102报告厅

摘要： Interactive mesh and image cutting refers to the process of extracting meaningful parts from an input mesh or image and involving a little user interaction to indicate user intention. It is an active research area in computer graphics and image processing in recent years. In this talk I will present some of our recent work in this area. Specifically, we propose a constrained random walks algorithm that adds constraints to the random walks procedure and thus allows for a variety of intuitive user inputs, and develop a unified framework for interactive mesh or image cutting that supports three typical user inputs and also their combinations: foreground/background seed inputs, soft constraint inputs, and hard constraint inputs. The varieties of user inputs make the cutting process intuitive, flexible and effective. Experimental results show that our algorithms are fast, reliable, and capable of producing good results reflecting user intention and geometric attributes.

L2 Projected Finite Element Methods for

Maxwell's Equations with Low Regularity Solution

演讲人：Professor Lin Ping

（University of Dundee, UK）

2010年12月10日（星期五）下午1：00-2：00

华东师大闵行校区数学楼102报告厅

摘要： In the talk we will present an element-local L2 projected finite element method to approximate the nonsmooth solution (not in H1) of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local L2 projectors are applied to both curl and div operators. The C0 linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in L2 norm is established uniformly in the mesh size. For the solution and its curl in H^r with r<1 we obtain an error bound O(h^r) in an energy norm. Numerical examples confirm the theoretical error bound. The idea is also applied to curl-div magnetostatic problem in multiply-connected Lipschitz polyhedrons. A desirable error bound is obtained as well.

On a Reaction-Diffusion System with Conservation of a Mass

演讲人：Professor Yoshihisa Morita
（Ryukoku University , Japan）
2010年11月4日（星期四）下午3:00—4:00
华东师大闵行校区数学系102报告厅

Abstract: We consider a reaction-diffusion system with conservation of a mass. In this system the stationary equations of the system are reduced to a scalar stationary equation with a nonlocal term. Namely, there is a one-to-one correspondence between equilibrium solutions of the system with a given mass and the scalar equation with the nonlocal term. As for the stability of equilibrium solutions, under some condition for the reaction terms, we show that if the scalar equation allows a nondegenerate stable solution, then the system has a stable solution corresponding to the scalar one, and vice versa. We also give a specific example having a spatially non-uniform periodic solution, which never arises in the scalar equation.

Some Recent Developments on Quantum Error-correcting Codes

演讲人：Professor Ling San
（Nanyang Technological University, Singapore ）
2010年10月21日（星期四）下午3:00—4:00
华东师大闵行校区数学系102报告厅

Abstract: Quantum error-correcting codes were introduced to protect quantum information from decoherence and quantum noise. Ever since their introduction in the mid 1990's, much progress has been made on their construction. A rather prevalent idea that has proved to be very useful is to construct quantum codes using classical error-correcting block codes. More recently, it was also established from experiments that, in many quantum systems, phase-flip errors happen more frequently than bit-flip errors. Asymmetric quantum error-correcting codes, which take advantage of this asymmetry, have been introduced as a result (as opposed to the more established symmetric ones which do not distinguish between these two types of errors). As in the case for classical error-correcting block codes, there are known bounds constraining the parameters of quantum codes (both symmetric and asymmetric cases). An interesting problem is to construct examples of ``best'' quantum codes, in the sense of attaining some of such bounds. One such bound is the quantum Singleton bound, and the codes whose parameters satisfy this bound are known as quantum maximum distance separable (MDS) codes.

In this talk, we will discuss some recent developments in the construction of quantum MDS codes, both symmetric and asymmetric, using purely mathematical approaches. No pre-requisite knowledge of error-correcting codes will be assumed. Essential ideas and facts in coding theory will be introduced.

Comparison of Rescaled Energy for a Supercritical Nonlinear Heat Equation
演讲人：Professor Hiroshi Matano
（University of Tokyo, Japan）
2010年10月18日（星期一）下午3:00—4:00
华东师大闵行校区数学系102报告厅

Abstract: In this talk I will discuss blow-up of radially symmetric solutions of the nonlinear heat equation with power nonlinearity, whose exponent is above the Sobolev critical exponent. Our first goal is to show that the so-called type II blow-up is a rather exceptional phenomenon, and almost all blow-ups are of type I. Our method is based on a combination of the zero-number principle and energy estimates. The second goal of my talk is to show that the constant positive solution attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. This means that the flat self-similar solution is most stable among all self-similar blow-up solutions. This leads to a sharp criterion for no blow-up. Interestingly, these two apparently different results are closely related to each other and can be derived from the same argument. This is joint work with Frank Merle.

1. Bulk Superconductivity for Magnetic Field Strengths

Close to the Second Critical Field

2. On a 1-dimensional Spectral Problem Related to

Surface Superconductivity

演讲人：Professor Soeren Fournais
   Aarhus University, 丹麦
1.  2010年9月16日（星期四）下午3:00—4:00
  2.  2010年9月21日（星期二）下午3:00—4:00
华东师大闵行校区数学系102报告厅

Abstract: 1. In the theory of superconductors of Type II in the Ginzburg-Landau model, there is a critical field called the second critical field where superconductivity goes from being a bulk to a boundary phenomenon. A precise mathematical definition is hard to give but one can observe this phenomenon in suitable limit situations. In this talk I will describe recent work both in 2 and 3 dimensions on the ground state energy density for the Ginzburg-Landau model in the Type II limit and show how the transition to boundary superconductivity is evident from the energy density. This is joint work with Ayman Kachmar.

2. We consider 2 dimensional superconductors in the Ginzburg-Landau model. In the regime where superconductivity of Type II superconductors is a boundary phenomenon it was proved by Pan that superconductivity is uniformly distributed in energy along the boundary. However, the calculation of the constant appearing in this energy density, i.e. the strength of the boundary superconductivity is only conjectures. In this talk we will describe how this non-linear problem can be reduced to the study of a 1-dimensional spectral problem and we will give some spectral results on this 1D problem. This is joint work with Bernard Helffer and Mikael Persson.

Chaotic Behavior for Periodically Forced Parabolic Equations

演讲人： Professor Kening Lu

Brigham Young University, USA

2010年8月6日（星期五）下午2：30--3：30

华东师大中山北路校区理科大楼A座1510室

Abstract: We prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

Ansatz Solutions to a Problem of Mean Curvature and Newtonian Potential

演讲人： Professor Xiaofeng Ren

The George Washington University

2010年7月16日 (星期五),下午3：10-- 4：10

华东师范大学闵行校区数学系102报告厅

Abstract：Pattern formation problems arise in many physical and biological systems as orderly outcomes of self-organization principles. Examples include animal coats, skin pigmentation, and morphological phases in block copolymers. Recent advances in singular perturbation theory and asymptotic analysis have made it possible to study these problems rigorously. In this talk I will discuss a central theme in the construction of various patterns as solutions to some well known PDE and geometric problems: how a single piece of structure built on the entire space can be used as an ansatz to produce a near periodic pattern on a bounded domain. We start with the simple disc and show how the spot pattern in morphogenesis and the cylindrical phase in diblock copolymers can be mathematically explained. More complex are the ring structure and the oval structure which can also be used to construct solutions on bounded domains. Finally we discuss the newly discovered smoke-ring structure and the toroidal tube structure in space. The results presented in this lecture come from joint works with Kang, Kolokolnikov, and Wei.

Kinetics Causing the Collapse of Patterns

演讲人： Professor Izumi Takagi

Tohoku University

2010年6月21日（星期一）下午1：30--2：30

华东师大闵行校区数学系102报告厅

Abstract. Gierer and Meinhardt proposed a reaction-diffusion system comprised of an activator and an inhibitor to model transplantation experiments on hydra. If the equations do not contain terms independent of the activator and the inhibitor, then it is numerically observed that some solutions of the initial-boundary value problem starting from almost constant initial data develop spatial inhomogeneity at first, then begin to oscillate, and eventually converge uniformly to zero. Since the equation has a term with the inhibitor in the denominator, this limit is not a stationary solution. This phenomenon, called the collapse of patterns, seems to be known among people doing computer simulations, but has never been reported in literature. In this talk we prove rigorously that under certain conditions patterns do collapse, and study what happens if we perturb the system slightly to understand the mechanism that causes the collapse of patterns.

Strauss conjecture for nontrapping obstacles

演讲人： Assist. Prof. Wang Chengbo

Johns Hopkins University

2010年6月18日（星期五）下午1：30--2：30

华东师大闵行校区数学系102报告厅

Abstract. In this talk, we discuss our recent work on the 2-dimensional Strauss conjecture for nontrapping obstacles. This is a joint work with H. Smith and C. Sogge. Recently, Hidano, Metcalfe, Smith, Sogge and Zhou proved the Strauss conjecture for nontrapping obstacles when the spatial dimension n equals 3 and 4. Their method is to prove abstract Strichartz estimates, including the |x|-weighted Strichartz estimates. In the Minkowski spacetime, the |x|-weighted Strichartz estimates (also from the work of Fang and Wang) can be utilized to prove the Strauss conjecture with n = 2, 3, 4. The reason that they can only prove the general results for n = 3, 4 is that the abstract Strichartz estimates are proved only for the case with regularity s in [(3-n)/2, (n-1)/2] (that is, s = 1/2 if n = 2). This restriction is essential for the general abstract Strichartz estimates.

In this work, we remedy this difficulty for n=2 by proving the generalized Strichartz estimates of the type L^q_t L^r_{|x|} L^2_\theta. The corresponding problem for n>4 are still open.

非线性泛函分析的若干方面

演讲人：郭大钧教授

山东大学数学系

2010年6月8日（星期二）下午3：00-4：00，

闵行校区数学系102报告厅

摘要: 简要介绍报告人五十多年科研工作的主要成果,包括三个方面: 第一,建立新的有使用价值的不动点定理(重点是范数形式的锥拉伸与锥压缩不动点定理); 第二,利用半序方法研究非紧非连续增算子,减算子和混合单调算子的不动点(以混合单调算子为例说明); 第三,利用拓扑方法,变分方法和半序方法研究各种非线性积分方程的解(包括Banach空间中非线性脉冲积分-微分方程的解).

下降流不变集方法及其应用

演讲人：孙经先教授

徐州师范大学数学系

2010年6月1日（星期二）下午2：00-3：00，

闵行校区数学系102报告厅

摘要: 下降流不变集方法把临界点的个数问题归结为下降流不变集的个数问题，给出了研究临界点个数的一个新思路，由此可以获得若干新的多临界点存在的定理，为研究具有变分结构的微分方程解的个数和变号解的存在性提供了新的工具。

研究生短课程:

Lectures on the Nodal and the Critical Sets

演讲人：Professor Fanghua Lin

（Courant Institute of Mathematical Sciences, New York University）

2010年5月24, 25, 26, 27, 28日，下午1:30-4:00,

闵行校区数学楼102报告厅

Contents

Part 1. A description of the subject and its relevance.
Part 2. Basic ideas for handling nodal sets.
Part 3. Basic ideas for handling critical sets.
Part 4. Applications.

研究生短课程: 椭圆方程基态解的动力系统方法

Dynamical Systems Methods for Ground State Problems

演讲人：Professor Russell Johnson

（Università di Firenze, Italy. 意大利佛罗伦萨大学）

2010年5月14, 17, 19日，下午1:00-4:00,

闵行校区数学楼102报告厅

课程简介: In recent years, several authors have used certain methods of dynamical systems theory to obtain results concerning the existence and multiplicity of radially symmetric positive solutions of various semilinear and nonlinear elliptic PDEs. Efforts along these lines were initiated by Johnson-Pan-Yi, and further advances have been effectuated by Battelli, Bamon, Flores, Franca and other mathematicians. We will discuss some of the background facts necessary to understand these methods, then analyze specific problems as time permits.

2009年研究生短课程与学术报告

Mini-Courses for Graduate Students and Seminar Lectures

研究生短课程: 调和映照的正则性

Regularity of Harmonic Maps

演讲人：Professor Robert Hardt

（Rice University）
2009年6月22, 23, 24日，下午1:30-3:30,

闵行校区数学楼102报告厅

课程简介: Harmonic maps between Riemannian manifolds are critical points for the Dirichlet energy and naturally generalize both geodesics and real-valued harmonic functions. Ideas concerning their regularity and singularity have proven useful for not only harmonic maps but for many other variational problems and applications (including liquid crystals and ferromagnetism). Lecture 1 will contain a summary, overview, and discussion of examples. Lecture 2 will discuss many basic ideas in one proof of the partial regularity for sphere-valued energy-minimizing maps. Lecture 3 will involve ideas used in the construction of many unusual regular and singular harmonic maps that are not energy-minimizing. Here there are several open questions.

Energy-law Preserving C^0 Finite Element Methods for

Simulation of Liquid Crystal and Multi-phase Flows

演讲人： Professor Lin Ping
（University of Dundee, UK）

2009年6月5日（星期五）下午2:40-3:40,

华东师大闵行校区数学系102室

Abstract: The liquid crystal (LC) flow model is a coupling between orientation (director field) of LC molecules and a flow field. The model may probably be one of simplest complex fluids and is very similar to an Allen-Cahn phase field model of multiphase flows if the orientation variable is replaced by a phase function. It is very much of multiscale feature due to a few large or small parameters involved in the model (e.g. the small penalty parameter for the unit length LC molecule or the small phase-change parameter, possibly large Reynolds number of the flow field, etc.). We propose a C^0 finite element formulation in space and a modified midpoint scheme in time which accurately preserves the inherent energy law of the model. We use C^0 elements because they are simpler than existing C^1 element and mixed element methods. We emphasize the energy law preservation because from the PDE analysis point of view the energy law is very important to correctly catch the evolution of singularities in the LC molecule orientation. In addition we will see numerical examples that the energy law preserving scheme performs better in the multiscale situation, for example, it works better for high Reynolds number flow fields and allows coarser grids to qualitatively catch the interface evolution of the two-phase flow. Finally we apply the same idea to a Cahn-Hilliard phase field model where the biharmonic operator is decomposed into two Laplacian operators. But we find that under a C^0 finite element setting non-physical oscillation near the interface occurs. From the viewpoint of differential algebraic equations this is because the modified midpoint scheme is not of stiff decay. The oscillation can be removed by doing only one step of a modified backward Euler scheme at the beginning time. A number of numerical examples demonstrate the case. We will mention possible application of the method to the superconductivity model as well.

研究生短课程:椭圆方程斜微商问题

Oblique Derivative Problems of Elliptic Equations

Professor Gary Lieberman

（Iowa State University）

2009年5月12,14,15,19,21,22日，下午1：00-4：00

闵行校区数学楼102报告厅

课程简介：在2008年短课程的基础上，介绍椭圆型偏微分方程斜微商问题正则性理论，包括Schauder估计与弱解的W^{2,p}估计。

漫谈素数

演讲人：董光昌教授

（浙江大学）

2009年5月1日下午3：30-4：30，

中山北路校区理科大楼1510室

On the fast reaction limit of a competition-diffusion system

Professor Danielle Hilhorst

（Universite de Paris-Sud 11, 巴黎第11大学）

2009年4月28日（星期二）下午3：00-4：00,

闵行数学楼102室

Abstract: We consider a competition-diffusion system for two competing species; the density of the first specie satisfies a parabolic equation whereas the second one either satisfies a parabolic equation or an ordinary differential equation. We show that the two species spatially segregate as the interspecific competition rate becomes large; the limit problem turns out to be a free boundary problem, which may have the form of a Stefan problem. We first present a convergence result as well as an error estimate. We then focus on the singular limit of the interspecific reaction term, which involves a measure located on the free boundary. This is joint work with S. Martin, M. Mimura and H. Ninomiya.

研究生短课程

On Schrodinger operators with magnetic field or periodic potential :

analysis of flux effects

主讲人：Professor Bernard Helffer

（Universite de Paris-Sud 11, 巴黎第11大学）

2009年3月24日，26日, 31日 , 四月2日，下午1：00-4：00

闵行校区数学楼102室

课程简介：In these lectures, we would like to analyze some spectral properties of Schrodinger operators in the case of a periodic potential or with a magnetic field. For the Schrodinger operator with a periodic potential we will discuss Floquet theory and Wannier functions, and for the Schrodinger operator with a magnetic field we discuss diamagnetism, Aharonov-Bohm effect, multiplicity of eigenvalues, and nodal sets.

Dynamics of fronts in a reaction-diffusion equation of multistable type

演讲人： Professor Ken-Ichi Nakamura

（University of Electro-Communications, Tokyo, Japan）
　　2009年3月20日（星期五）下午3:10-4:10，

闵行数学楼102

Abstract: In this talk, we consider the dynamics of fronts for some reaction-diffusion equations having multi-stable reaction term. A typical example of such equations is the overdamped sine-Gordon equation. We will discuss generation of multiple fronts (with stairs-like shape), generation of metastable fronts and slow motion of fronts.

2008年研究生短课程与学术报告

Mini-Courses for Graduate Students and Seminar Lectures

Stability of the normal state of superconductors in the presence

of electric currents with or without magnetic fields

演讲人：Professor Yaniv Almog

（Louisiana State University）
　　2008年12月25日（星期四）下午3：00-4：00，闵行数学楼102

Abstract: A review of recent results on the stability of the normal state in the presence of electric currents would first be presented. Then, the effect of applied and induced magnetic fields on solutions in unbounded domains will be presented. The latter results are part of a work in progress which is done in collaboration with Bernard Helffer from Orsay.

PDE methods for weak KAM theory
　　

演讲人：Professor L. C. Evans

（University of California Berkeley University）
　　2008年12月20日（星期六）下午3：00-4：00，中山北路理科大楼A1510

Mathematical Analysis and Numerical Simulation of Bose-Einstein Condensation

演讲人： Professor Weizhu Bao

(National University of Singapore)

2008年12月14日（星期四）下午3：00-4：00，闵行数学楼102

Abstract: In this talk, I review the mathematical results of the dynamics of Bose-Einstein condensate (BEC) and present some efficient and stable numerical methods to compute ground states and dynamics of BEC. As preparatory steps, we take the 3D Gross-Pitaevskii equation (GPE) with an angular momentum rotation, scale it to obtain a four-parameter model and show how to reduce it to 2D GPE in certain limiting regimes. Then we study numerically and asymptotically the ground states, excited states and quantized vortex states as well as their energy and chemical potential diagram in rotating BEC. Some very interesting numerical results are observed. Finally, we study numerically stability and interaction of quantized vortices in rotating BEC. Some interesting interaction patterns will be reported.

Positive solutions of an elliptic equation with negative exponent: stability and critical power

演讲人： Professor Yihong Du

(University of New England, Australia)

2008年11月13日（星期四）下午3：00-4：00，闵行数学楼102

Preservation of ergodicity in nonlinear evolution equations and its applications

演讲人： Professor Hiroshi Matano

(University of Tokyo)

2008年11月6日（星期四）下午3：00-4：00，闵行数学楼102

Abstract: A recurrent function is called ergodic if it has certain averaging property. This concept is wider than and is often more natural than almost periodicity when one studies spatially heterogeneous behavior of solutions of nonlinear evolution equations. In this talk, I first present a rather simple general principle that states that spatial ergodicity in the equation or in the initial data is inherited by the solutions. I will then apply this principle to front propagation in nonlinear diffusion equations and show that:

(1) travelling waves in spatially ergodic media have well-defined average speed;

(2) planar fronts in the spatially homogeneous Allen-Cahn equation are asymptotically stable under spatially ergodic perturbations.

Front dynamics and entire solutions to reaction-diffusion equations

演讲人：Professor Yoshihisa Morita

(Ryukoku University, Japan, morita@rins.ryukoku.ac.jp)

2008年10月23日（星期四）下午3：00-4：00，闵行数学楼102

Abstract: It is interesting to study propagation phenomena modeled by reaction-diffusion equations in various fields of materials science, biology and life science. Corresponding to a wave propagation, the equations allow a traveling wave that has a constant profile and a constant speed. Here we consider a simple model equation, that is a scalar reaction-diffusion equation of one-space dimension $u_t=u_{xx}+f(u)$ with a bistable condition. It is known that the equation allows 6 types of monotone traveling waves (traveling fronts), counting the solutions obtained by the reflection in the space. In addition to the traveling fronts, we can also observe interacting behavior of two fronts. In this talk we show the existence of entire solutions which behaves in the way two traveling fronts are coming or diverging. Here an entire solution is meant by a classical solution defined for all $x$ and $t$. Although an equilibrium solution and a traveling wave solution are examples of entire solutions, our entire solutions are different from those and exhibit characteristic behaviors in the front dynamics.

研究生短课程《椭圆与抛物方程斜微商问题》

Oblique Derivative Problems of Elliptic and Parabolic Equations
主讲人： Professor Gary Lieberman

（Iowa State University）

时间： 2008年10月8,9,10,14,15,17日，下午1：00-4：00
地点：闵行校区数学楼102报告厅
课程简介：介绍椭圆型与抛物型偏微分方程斜微商问题基本理论，包括解的正则性理论与先验估计。

参考书：D.Gilbarg and N.Trudinger, Elliptic Partial Differential Equations of Second Order.
G.Lieberman, Second Order Parabolic Differential Equations.

On elliptic equations with strongly singular lower order terms

演讲人： Professor Gary Lieberman

(Iowa State University)

2008年10月16日（星期四）下午3：00-4：00，闵行数学楼102

Traveling wave solutions of lattice dynamical systems

演讲人： Professor Guo Jong-Sheng

(National Taiwan Normal University)

2008年7月15日（星期二）10：00-11：00，理科大楼510

研究生短课程《自由边界问题》

Free Boundary Problems

主讲人: Professor Fanghua Lin

（New York University）

（1） 6月4日星期三，下午2：00-4：00

地点：中山北路校区理科大楼A-1414

（2） 6月5日星期四, 下午2：00-4：00

地点：中山北路校区理科大楼A-510

（3） 6月6日星期五，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（4） 6月11日星期三, 上午10：00-12：00

地点：闵行校区数学楼102

（5） 6月17日星期二，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（6） 6月18日星期三，下午2：00-4：00

地点：中山北路校区理科大楼A-1414

（7） 6月19日星期四，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（8） 6月20日星期五，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（9） 6月23日星期一，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（10）6月24日星期二，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（11）6月25日星期三，下午2：00-4：00

地点：中山北路校区理科大楼A-510

（12）6月26日星期四，下午2：00-4：00

地点：中山北路校区理科大楼A-1414

研究生短课程《几何测度论》

Geometric Measure Theory

主讲人： Professor Thierry de Pauw

(Univ. Catholique de Louvain)

(1) 6月10日星期二，下午1：30--3：30，闵行校区教三-413

(2) 6月11日星期三，下午1：30--3：30，闵行校区教三-406

(3) 6月12日星期四，下午1：30--3：30，闵行校区教三-406

(4) 6月13日星期五，下午1：30--3：30，闵行校区教三-406

(5) 6月16日星期一，下午1：30--3：30，闵行校区数学楼126

6月14日下午, 地点：中山北路校区理科大楼A-510

2：00-3：00，Prof. Fanghua Lin，Hopf Invariants and Knotted Solitons in High Dimensions

3：10-4：10，Prof. Thierry de Pauw，Theory of Minimal Surfaces—Some Recent Developments

In Search of Mathematical Truth

演讲人： Professor Chong Chi Tai

(National University of Singapore)

2008年5月26日（星期一）下午2：30-3：30，闵行教4#-325

Abstract: The nature of mathematical truth has been the subject of intensive research from both the mathematical and philosophical points of view for more than a hundred years. Despite significant advances, it is not clear that we are closer to the ultimate answer than when the subject was first raised. This talk surveys the milestones that led to the current state of knowledge and discusses the key issues and questions that are of major interest.

On Perfect Nonlinear Functions

演讲人：Professor Ling San

(Nanyang Technological University, Singapore)

2008-5-21（周三）下午3：00-4：00，闵行教4# 303

研究生短课程《薛定谔算子半经典分析》

Introduction to semi-classical analysis for Schroedinger operators
主讲人：Professor Bernard Helffer

（Universite de Paris-Sud 11, 巴黎第11大学）

4月21日（周一）下午1：00-4：00，闵行4＃423室
4月22日（周二）下午1：00-4：00，闵行3＃228室
4月23日（周三）下午1：00-4：00，闵行3＃228室
4月24日（周四）下午1：00-4：00，闵行3＃224室
4月25日 (周五）下午1：00-4：00，闵行3＃406室
4月28日（周一）下午1：00-4：00，闵行4＃423室

课程简介：

The main object is to introduce the students to the techniques permitting to measure the tunneling effect in Quantum mechanics in the limit when the Planck constant h tends to 0.

After some general introduction in semi-classical analysis, we will successively present in order to analyze the lowest eigenvalues of a Schroedinger operator: the harmonic approximation, the Agmon estimates and finally the analysis of the tunneling : we will concentrate the applications to the double well problem, the periodic problem and the Aharonov Bohm effect.Very shortly we can conclude by presenting the analogous results occuring in the semi-classical analysis of Schrodinger operators with magnetic fields.

On Nodal domains and spectral minimal partitions

演讲人：Professor Bernard Helffer

（Universite de Paris-Sud 11, 巴黎第11大学）

2008年4月29日（星期二）下午2：00-3：00，

闵行3＃228室

Speed of propagation of traveling fronts in periodic media governed by a bistable nonlinearity

演讲人Professor Ken-Ichi Nakamura

（University of Electro-Communications, Tokyo）

2008年2月21日（星期四）下午3：00-4：00，

理科大楼A座1510室

A characterization of optimal multifunctional composite designs

(多功能复合材料的最优设计)

演讲人： Dr. Luis Silvestre

(Courant Institute of Mathematical Sciences, New York University)

2007年6月5日（星期二）下午4：00-5：00

华东师大理科大楼A座 1510室

Abstract: We will discuss the problem of optimal design of a two-phase composite material to maximize the sum of thermal and electric conductivity. There was some numerical evidence that the optimal design was related to a periodic minimal surface. We obtain a characterization of an optimal configuration in terms of a free boundary type problem and show what the free boundary cannot be such minimal surface. In the process of obtaining this result, we provide a new proof of the relevant case of Bergman’s cross-property bound. With the same idea a simple new proof of the Hashin-Strikman bounds can also be obtained.

Superconductivity Near Hc2

演讲人: Professor Yaniv Almog

(Louisiana State University)

2007年5月25日（星期五）下午3：30-4：30

华东师大理科大楼A座1406 室

摘要： While precise definition of Hc2 is currently unavailable in the literature it is commonly perceived as the applied field value at which the superconductivity order parameter appears in the bulk of the sample. For larger values of the applied magnetic field it is well known that the order parameter is concentrated near the boundary. The existence of such a field has been conjectured by Physicists more than 40 years ago, but so far only a few mathematical results exist in the literature. In this talk I’ll review these results (some of them are my own) and relate them to the well-known work of Abrikosov, for which he won the Nobel Prize in Physics in 2003.

A Braid Group Method for Blow-up in Nonlinear Heat Equations

演讲人： Professor Hiroshi Matano

(东京大学数学系)

2007年5月23日（星期三）下午4：00-5：00

华东师大理科大楼A座510室。

Abstract: In this talk I will present an intriguing application of the braid group theory to the study of blow-up in a nonlinear heat equation $u_t = \Delta u + u^p$, where $p$ is supercritical in the Sobolev sense. The goal is to determine all the type II blow-up rates by analyzing the topological properties of certain braids.

International Conference on Mathematical Theory of

Superconductivity and Liquid Crystals

超导与液晶的数学理论国际会议

May 14-18,2007 ECNU, Shanghai

http://www.math.East China Normal University.edu.cn/SCLC-Conf

Conference Venue: ECNU, Science Building, Room A504

Scientific Committee:

Haim Brezis, Shuxing Chen, Weiyue Ding, Yoshikazu Giga, Bolin Guo, Robert Hardt,Bernard Helffer, Jiaxing Hong, Lishang Jiang, Daqian Li, Fanghua Lin (Chair), W.M. Ni, Xingbin Pan

Organizing Committee:

Bernard Helffer (Chair), Xingbin Pan, Xueping Wang, Danping Yang, Feng Zhou (Co-Chair)