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 cutting-edge 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)

Invited Speakers:

1. Denis Kriventsov (Rutgers University)

2. Fanghua Lin (New York Uniersity)

3. Joaquim Serra (ETH-Zurich),

4. Yannick Sire (John Hopkins University),

5. Kelei Wang (Wuhan University)

6. Jun-Cheng Wei (University of British Columbia)

Introduction:

Denis Kriventsov:

Title: Regularity of Free Boundaries via Linearization

Abstract: While the structure of free boundaries in Bernoulli and obstacle-type problems is potentially quite complicated, the fact remains that at generic points, one expects that the interface is smooth and well-approximated by some simple one-dimensional 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 Ginzburg-Landau system and the fractional Allen-Cahn 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 Ginzburg-Landau system

6- The fractional Allen-Cahn 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 well-known 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 Allen-Cahn equation, established recently with Juncheng Wei. The main foucus of the lectures will be devoted to the derivation of Toda system from Allen-Cahn 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 Allen-Cahn equation, a proof of the multiplicity one conjecture of Marques and Neves by Chodosh and Mantoulidis using Allen-Cahn approximation.

Juncheng wei:

Title: Parabolic gluing method and finite time singularity for two-dimensional nematic liquid crystal flow

Abstract: I will introduce recently developed parabolic gluing methods and its application to construction of finite time blow-up to two-dimensional nematic liquid crystals flows introduced by C. Liu and F. Lin.

TBA..

Schedule:

注册：14:00-20:00, May 19, 2019, Lobby in Zijingang International Hotel (紫金港国际饭店)

Date	Time	Speakers
May 20 (Monday)	9:00-10:00& 10:20-11:20	Yannick Sire
		Free
May 21 (Tuesday)	9:00-10:00& 10:20-11:20	Yannick Sire
	14:00-15:20 & 15:40-17:00	Kelei Wang
May 22 (Wednesday)	9:00-10:00& 10:20-11:20	Yannick Sire
	14:00-15:20 & 15:40-17:00	Kelei Wang
May 23 (Thursday)	9:00-10:00& 10:20-11:20	Yannick Sire
	14:00-15:20 & 15:40-17:00	Kelei Wang
May 24 (Friday)	9:00-10:00& 10:20-11:20	Juncheng Wei
	14:00-17:00 workshop
May25(Saturday)	9:00-11:30 workshop
	14:00-17:00 workshop
May 27 (Monday)	9:00-10:00& 10:20-11:20	Joaquim Serra
	14:00-15:00 & 15:20-16:20	Denis Kriventsov
May 28 (Tuesday)	9:00-10:00& 10:20-11:20	Joaquim Serra
	14:00-15:00 & 15:20-16:20	Denis Kriventsov
May 29 (Wednesday)	9:00-10:00& 10:20-11:20	Joaquim Serra
	14:00-15:00 & 15:20-16:20	Denis Kriventsov
May 30 (Thursday)	9:00-10:00& 10:20-11:20	Joaquim Serra
	14:00-15:00 & 15:20-16:20	Denis Kriventsov
May 31 (Friday)	9:00-10:00& 10:20-11:20	Fanghua Lin

Hotel Information:

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)

Contact:

Email: APDES2019@163.com

Sponsor:

Zhejiang University

University of Science and Technology of China

Tian-Yuan Foundation of NSFC

NSFC

Conference Schedule for Advanced Seminar in PDE