Advanced PDE Seminars
“偏微分方程前沿讨论班”旨在为国内偏微分方程领域青年教师和学生开设学科前沿课程，促进彼此间的交流和合作，已成功举办三届。前三届分别在浙江大学，中国科学技术大学，中科院数学所举办。第四届“偏微分方程前沿讨论班”将于2019年5月19日6月1日在浙江大学举办，这些课程将主要聚焦于椭圆和抛物型方程，与几何分析也联系密切，预计有80位青年教师和学生参加。
Advanced PDE Seminars aims to provide cuttingedge courses in partial differential equations for young scholars and students in China, and promote exchanges and cooperation between them. The Seminars have been successfully held for three years in Zhejiang University（Hangzhou, 2016）, University of Science and Technology of China(Hefei,2015) , Academy of Mathematics and Systems Science , Chinese Academy of Sciences (Beijing, 2014) respectively. The fourth Advanced PDE Seminars will be held in the School of Mathematical Sciences, Zhejiang University from May 19th to June 1st, 2019.The courses will focus on elliptic and parabolic equations, and have a strong favor of "geometric analysis". About 80 young scholars and students will attend the seminar.
Place:
Lecture hall in Institute of Advanced Mathematics, East 7 Building, Zijingang Campus, Zhejiang University, Hangzhou.
(浙江大学紫金港校区东7数学高等研究院报告厅)
Topics:
Nonlocal Elliptic and Parabolic Equations and Free Boundary Problems.
Committee:
Fanghua Lin (New York Uniersity)
Xinan Ma (University of Science and Technology of China)
Weimin Sheng (Zhejiang Uniersity)
Wei Wang (Zhejiang Uniersity)
Ting Zhang (Zhejiang Uniersity)
1. Denis Kriventsov (Rutgers University)
2. Fanghua Lin (New York Uniersity)
3. Joaquim Serra (ETHZurich),
4. Yannick Sire (John Hopkins University),
5. Kelei Wang (Wuhan University)
6. JunCheng Wei (University of British Columbia)
Denis Kriventsov:
Title: Regularity of Free Boundaries via Linearization
Abstract: While the structure of free boundaries in Bernoulli and obstacletype problems is potentially quite complicated, the fact remains that at generic points, one expects that the interface is smooth and wellapproximated by some simple onedimensional profile. An observation of De Giorgi, in his work on minimal surfaces, was that this is a great setup for a compactness argument which essentially reduces it to studying a linearized problem for smooth perturbations of the tangent object. We will discuss how the same insight can be applied to Bernoulli problems, following a work of De Silva. Time permitting, we will also consider some further recent extensions by the speaker and F. Lin, other free boundary problems which can be tackled, and the case of the obstacle problem.
Fanghua Lin:
Title: Liquid Crystal Droplet and its associated configuration of orientation.
Yannick Sire:
Title:Asymptotics for nonlocal geometric equations and applications
Abstract:The aim of the lectures is to develop tools in Geometric Measure Theory and calculus of variations to study geometric problem which are of nonlocal nature such as harmonic maps with free boundary and minimal surfaces with free boundary. We will investigate mainly two problems in singular perturbation theory: the fractional GinzburgLandau system and the fractional AllenCahn equation. Each of those exhibit nonlocal phenomena and new approaches have to be developed to deal with the limit.
1 Basic theory of standard Minimal Surfaces
2 Basic theory of standard Harmonic maps
3 Fractional harmonic maps and their extension as harmonic maps with free boundary
4 Nonlocal minimal surfaces
5 The half GinzburgLandau system
6 The fractional AllenCahn equation
7 Elements of parabolic theory
Joaquim Serra:
Title: THE OBSTACLE PROBLEM: REGULARITY OF THE FREE BOUNDARY AND ANALYSIS OF SINGULARITIES
Abstract: The classical obstacle problem can be seen as the paradigm of free boundary problems. It is equivalent, after certain transformations, to other wellknown free boundary problems such as the Stefan problem. We will give an introduction to the regularity theory for the free boundary of the obstacle problem. We will revisit the classical theory (with some new proofs) and also explain some recent exciting developments on the analysis of singularities.
Kelei Wang:
Title: Second order estimates on transition layers
Abstract: I will discuss a second order regularity theory on level sets of solutions to the singularly perturbed AllenCahn equation, established recently with Juncheng Wei. The main foucus of the lectures will be devoted to the derivation of Toda system from AllenCahn equation, by using the infinite dimensional Lyapunov–Schmidt reduction method devloped by M. del Pino, M. Kowalczyk and Juncheng Wei. If time permits, I will also discuss some applications of this second order regularity theory, including the classification of finite Morse index solutions of AllenCahn equation, a proof of the multiplicity one conjecture of Marques and Neves by Chodosh and Mantoulidis using AllenCahn approximation.
Juncheng wei:
Title: Parabolic gluing method and finite time singularity for twodimensional nematic liquid crystal flow
Abstract: I will introduce recently developed parabolic gluing methods and its application to construction of finite time blowup to twodimensional nematic liquid crystals flows introduced by C. Liu and F. Lin.
TBA..
Schedule:
注册：14:0020:00, May 19, 2019, Lobby in Zijingang International Hotel (紫金港国际饭店)
Date 
Time 
Speakers 
May 20 (Monday) 
9:0010:00& 10:2011:20 
Yannick Sire 

Free 

May 21 (Tuesday) 
9:0010:00& 10:2011:20 
Yannick Sire 
14:0015:20 & 15:4017:00 
Kelei Wang 

May 22 (Wednesday) 
9:0010:00& 10:2011:20 
Yannick Sire 
14:0015:20 & 15:4017:00 
Kelei Wang 

May 23 (Thursday) 
9:0010:00& 10:2011:20 
Yannick Sire 
14:0015:20 & 15:4017:00 
Kelei Wang 

May 24 (Friday) 
9:0010:00& 10:2011:20 
Juncheng Wei 
14:0017:00 workshop 


May25(Saturday) 
9:0011:30 workshop 

14:0017:00 workshop 


May 27 (Monday) 
9:0010:00& 10:2011:20 
Joaquim Serra 
14:0015:00 & 15:2016:20 
Denis Kriventsov 

May 28 (Tuesday) 
9:0010:00& 10:2011:20 
Joaquim Serra 
14:0015:00 & 15:2016:20 
Denis Kriventsov 

May 29 (Wednesday) 
9:0010:00& 10:2011:20 
Joaquim Serra 
14:0015:00 & 15:2016:20 
Denis Kriventsov 

9:0010:00& 10:2011:20 
Joaquim Serra 

14:0015:00 & 15:2016:20 
Denis Kriventsov 

May 31 (Friday) 
9:0010:00& 10:2011:20 
Fanghua Lin 
Zijingang International Hotel, 796 Shenhua Road, Hangzhou
(紫金港国际饭店, 申花路796号)
http://hzzijinganginternationalhotel.vip.lechengol.com/
We could support 40 young researchers and Ph.D students' accommodations during the period, (2 person share one Standard Room)
Email: APDES2019@163.com
Zhejiang University
University of Science and Technology of China
TianYuan Foundation of NSFC
NSFC
Conference Schedule for Advanced Seminar in PDE
May 24, Friday, Afternoon Session 

14:00 – 14:10 
Opening: Profesor Fanghua Lin 
14:10 – 14:40 
Chuanqiang Chen: Smooth solutions to the $L_p$Dual Minkowski problem 
14:40 – 15:10 
Yuning Liu: Sharp interface limit of a phase field model for elastic bending energy 
15:10– 15:30 
Tea break 
15:30– 16:00 
Bin Deng：The Neumann problem for a class of fully nonlinear elliptic partial differential equations 
16:00– 16:30 
Caihong Yi：A class anisotropic curvature flows and $L_p$ Minkowski type problems 
16:30– 17:00 
Jiaxiang Wang：Some Results in complex MongeAmpere equations and Kaehler geometry 

Dinner 
May 25, Saturday, Morning Session 

9:00 – 9:50 
Tianling Jin：Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities 
9:50 – 10:20 
Yong Liu：Saddle solution of the AllenCahn equation in dimension 8 
10:20 – 10:40 
Tea break 
10:40 – 11:10 
Haigang Li：Babuska Problem in Composite Materials 
11:10– 11:40 
Wen Yang：Sharp estimates for solutions of Mean Field equation with collapsing singularities 

Lunch 
May 25, Saturday, Afternoon Session 

14:00 – 14:30 
Weiwei Ao: On the bubbling solutions of the MaxwellChernSimons model on flat torus 
14:30 – 15:00 
Yuan Cai: Global Wellposedness for Incompressible MHD 
15:00 – 15:20 
Tea break 
15:20 – 15:50 
Yiming Su：Construction of blowup and multisolitary solutions to the mass critical nonlinear Schrodinger equation 
15:50– 16:10 
Wei Wang：Some analysis results on profiles of interfaces and defects in liquid crystals 

Dinner 
On the bubbling solutions of the MaxwellChernSimons model on flat torus
Weiwei Ao(敖微微) < wwao@whu.edu.cn>
Wuhan University
Abstract: We consider the MaxwellChernSimons model on flat torus. First we consider the ChernSimons limit case and derive a BrezisMerle type alternative results and also construct solutions with concentration phenomena. This is joint work with Y. Lee and O. Kwon.
Smooth solutions to the $L_p$Dual Minkowski problem
Chuanqiang Chen(陈传强)
Zhejiang University of Technology
Abstract: In this talk, we consider the $L_p$dual Minkowski problem and some related problems. By studying the a priori estimates, curvature flows, we establish the existence theorem of the smooth solutions. This is a recent joint work with Yong Huang, and Yiming Zhao.
Global Wellposedness for Incompressible MHD
Yuan Cai(蔡圆)
The Hong Kong University of Science and Technology
Abstract:We study the Cauchy problem of the incompressible magnetohydrodynamic systems with or without viscosity. Under the assumption that the initial velocity field and the displacement of the initial magnetic field from a nonzero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally wellposed for all time and all space dimension n>1. Such a result holds true uniformly in nonnegative viscosity parameter.
The Neumann problem for a class of fully nonlinear elliptic partial differential equations
Bin Deng(邓斌) <bingomat@mail.ustc.edu.cn>
University of Science and Technology of China
Abstract: In this talk, I will study the Neumann problem for a class of fullly nonlinear elliptic equations, basically an extension of kHessian equations. Using maximum principle and suitable barrier function, we established a global $C^2$ estimates to the Neumann problem. By the method of continuity, we obtained the existence theorem of $k$admissible solutions of the Neumann problems.
Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities
Tianling Jin(金天灵)
The Hong Kong University of Science and Technology
Abstract: We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of CaffarelliGidasSpruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup*inf type Harnack inequality of Schoen for integral equations.
Babuska Problem in Composite Materials
Haigang Li(李海刚)
Beijing Normal University
Abstract: In highcontrast composite materials, the stress concentration is a common phenomenon when inclusions are close to touch. It always causes damage initiation. This problem was proposed mathematically by Ivo Babuska, concerning the system of linear elasticity, modeled by a class of second order elliptic systems of divergence form with discontinuous coefficients. I will first review some of our results on upper bound estimates by developing an iteration technique with respect to the energy integral to overcome the difficulty from the lack of maximal principle for elliptic systems, then present two very recent results of myself on lower bound estimates and asymptotics of the gradients to show that the blowup rates are actually optimal in dimensions two and three.
Saddle solution of the AllenCahn equation in dimension 8
Yong Liu(刘勇) <yliumath@ustc.edu.cn>
University of Science and Technology of China
Abstract: Simons' cone is area minimizing in dimesion 8. A corresponding conjecture for the AllenCahn equation is that the saddle solution is stable in dimension 8. Towarding this conjecture, in this talk, we discuss several qualitative properties of the saddle solution.
Sharp interface limit of a phase field model for elastic bending energy
Yuning Liu(刘豫宁)
New York Univeristy in Shanghai
Abstract: We investigate the phasefield approximation of the Willmore flow. This is a fourth order diffusion equation with a parameter $\epsilon>0$ that is proportional to the thickness of the diffuse interface. We show rigorously that for wellprepared initial data, as ε trends to zero the levelset of the solution will converge to motion by Willmore flow before the singularity of the latter occurs. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step and also the major contribution of this work is to show a spectrum condition of the linearized operator at the optimal profile. This is a fourthorder operator written as the sum of the squared AllenCahn operator and a singular linear perturbation. Our approach employs the spectrum decomposition with respect to the optimal profile, and such a decomposition brings in integrals of order up to $\epsilon^{4}$. The controls of these integrals make use of the separationofvariables properties of the asymptotic expansions, and the cancellation properties of the related integrals involving the optimal profile. This is a joint work with Mingwen Fei.
Construction of blowup and multisolitary solutions to the mass critical nonlinear Schrodinger equation
Yiming Su(苏一鸣)
Zhejiang University of Technology
Abstract: In this talk, we are concerned with the long time dynamics of the nonlinear Schrodinger equation in the mass critical setting. By duality we reduce the construction of multisolitary waves to the construction of blowup solutions at multiple points, which improves the similar result in the previous work of Frank Merle.
Some analysis results on profiles of interfaces and defects in liquid crystals
Wei Wang(王伟)
Zhejiang University
Abstract: We consider the profile solutions describing the isotropicnematic interface and 2D point defects under the framework of Qtensor theory. We will mainly talk about existences, stabilities and uniqueness of these profile solutions.
Sharp estimates for solutions of Mean Field equation with collapsing singularities.
Wen Yang(杨文) <math.yangwen@gmail.com>
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences
Abstract: In the seminar work, BrezisMerle, LiShafrir, BartolucciTarantello showed that any sequence of blow up solutions for (singular) Mean field equations must exhibit a "mass concentration" property. In this talk, I will show this phenomenon might not occur in general by analyzing the blow up solution of the Mean field equation with collapsing singularities. Among other facts, I will present that in certain situations, the collapsing rate of the singularities can be used to describe the blow up rate.
主办单位：浙江大学数学科学学院，浙江大学数学高等研究院，中国科学技术大学
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