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## Advanced PDE Seminar May 19-- June 1st, 2019

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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.

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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¨CSchmidt 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:

×¢²á£º14:00-20:00, May 19, 2019, Lobby in Zijingang International Hotel (×Ï½ð¸Û¹ú¼Ê·¹µê)

 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

(×Ï½ð¸Û¹ú¼Ê·¹µê, Éê»¨Â·796ºÅ)

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

Zhejiang University

University of Science and Technology of China

Tian-Yuan Foundation of NSFC

NSFC

Conference Schedule for Advanced Seminar in PDE

 May 24, Friday, Afternoon Session 14:00 ¨C 14:10 Opening: Profesor Fanghua Lin 14:10 ¨C 14:40 Chuanqiang Chen: Smooth solutions to the $L_p$-Dual Minkowski problem 14:40 ¨C 15:10 Yuning Liu: Sharp interface limit of a phase field model for elastic bending energy 15:10¨C 15:30 Tea break 15:30¨C 16:00 Bin Deng£ºThe Neumann problem for a class of fully nonlinear elliptic partial differential equations 16:00¨C 16:30 Caihong Yi£ºA class anisotropic curvature flows and $L_p$ Minkowski type problems 16:30¨C 17:00 Jiaxiang Wang£ºSome Results in complex Monge-Ampere equations and Kaehler geometry Dinner

 May 25, Saturday, Morning Session 9:00 ¨C 9:50 Tianling Jin£ºAsymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities 9:50 ¨C 10:20 Yong Liu£ºSaddle solution of the Allen-Cahn equation in dimension 8 10:20 ¨C 10:40 Tea break 10:40 ¨C 11:10 Haigang Li£ºBabuska Problem in Composite Materials 11:10¨C 11:40 Wen Yang£ºSharp estimates for solutions of Mean Field equation with collapsing singularities Lunch

 May 25, Saturday, Afternoon Session 14:00 ¨C 14:30 Weiwei Ao: On the bubbling solutions of the Maxwell-Chern-Simons model on flat torus 14:30 ¨C 15:00 Yuan Cai: Global Well-posedness for Incompressible MHD 15:00 ¨C 15:20 Tea break 15:20 ¨C 15:50 Yiming Su£ºConstruction of blow-up and multi-solitary solutions to the mass critical nonlinear Schrodinger equation 15:50¨C 16:10 Wei Wang£ºSome analysis results on profiles of interfaces and defects in liquid crystals Dinner

Titles and Abstracts

On the bubbling solutions of the Maxwell-Chern-Simons model on flat torus

Weiwei Ao(°½Î¢Î¢) < wwao@whu.edu.cn>

Wuhan University

Abstract: We consider the Maxwell-Chern-Simons model on flat torus. First we consider the Chern-Simons limit case and derive a Brezis-Merle 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(³Â´«Ç¿

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 Well-posedness for Incompressible MHD

Yuan Cai(²ÌÔ²)

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 non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed 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(µË±ó) <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 k-Hessian 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(½ðÌìÁé)

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 Caffarelli-Gidas-Spruck 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(Àîº£¸Õ)

Beijing Normal University

Abstract: In high-contrast 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 blow-up rates are actually optimal in dimensions two and three.

Saddle solution of the Allen-Cahn equation in dimension 8

Yong Liu(ÁõÓÂ) <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 Allen-Cahn 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(ÁõÔ¥Äþ

New York Univeristy in Shanghai

Abstract: We investigate the phase-field 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 well-prepared initial data, as ¦Å trends to zero the level-set 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 fourth-order operator written as the sum of the squared Allen-Cahn 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 separation-of-variables 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 blow-up and multi-solitary solutions to the mass critical nonlinear Schrodinger equation

Yiming Su(ËÕÒ»Ãù)

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 multi-solitary waves to the construction of blow-up 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(ÍõÎ°)

Zhejiang University

Abstract: We consider the profile solutions describing the isotropic-nematic interface and 2-D point defects under the framework of Q-tensor 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(ÑîÎÄ) <math.yangwen@gmail.com>

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences

Abstract: In the seminar work, Brezis-Merle, Li-Shafrir, Bartolucci-Tarantello 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.

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