Progress in Nonlinear Partial Differential Equations
(PNPDE2012)
JUNE 4-8, 2012
ZHEJIANG UNIVERSITY
HANGZHOU, CHINA
Organized by

Department of Mathematics, Zhejiang University
Sponsored by

Zhejiang University, through Qiu-Shi Jiang Zuo Professorship
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CONTENTS
SCIENTIFIC and ORGANIZING COMMITTEE
PROGRAM
ABSTRACT
INVITED SPEAKERS
GNERAL INFORMATION
2
3
4-6
7-15
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17-18
SCIENTIFIC and ORGANIZING COMMITTEE
Fanghua Lin (Chair)
New York University
USA
Gang Bao
Zhejiang University
China
H. Brestycki
University of Chicago
USA
L. Caffarelli
University of Texas at Austin
USA
Daoyuan Fang
Zhejiang University
China
Song Li
Zhejiang University
China
Yanyan Li
University of Chicago
USA
Chengbo Wang
Zhejiang University
China
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PROGRAM
Day 1- June 4th, 2012, Monday
8:45--9:00 AM
Opening Remarks
Chair
Daoyuan Fang
9:00--10:30 AM
Chang-Shou Lin
Elliptic integrable system: A case study
10:30--11:00 AM
Coffee-Tea Break
11:00--11:45 AM
Xavier Cabre
Elliptic problems with axial symmetry: antisymmetry and isoperimetry
LUNCH
Chair
Changyou Wang
2:00--2:45 PM
Yanyan Li
Some analytic aspects of conformally invariant fully nonlinear
equations
2:45--3:15 PM
Coffee-Tea Break
3:15--4:00 PM
Zhongwei Shen
Convergence Rates in Periodic Homogenization
4:00--4:15 PM
Short break
4:15--5:00 PM
Guozhen Lu
A new approach to sharp Moser-Trudinger and Adams inequalities in
unbounded domains: A rearrangement-free method
Day 2-June 5th, 2012, Tuesday
Chair
Chang-Shou Lin
9:00--10:30 AM
Luis Caffarelli
A counter-example in 3D and homogenization of mean curvature mot
ions in 2D
10:30--11:00 AM
Coffee-Tea Break
11:00--11:45 AM
Lihe Wang
Boundary regularity for quasiconvex functional
LUNCH
Chair
Xavier Cabre
2:00--2:45 PM
Irene Gamba
Solutions to the Boltzmann transport equation for soft potentials
2:45--3:15 PM
Coffee-Tea Break
3:15--4:00 PM
Luis Silvestre
On the continuity (or discontinuity) of solutions of
drift-diffusion equations
4:00--4:15 PM
Short break
4:15--5:00 PM
Yuan Lou
Evolution of dispersal in heterogeneous landscapes
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6:00 PM
Banquet
5
Day 3-June 6th, 2012, Wednesday
Chair
Luis Caffarelli
8:45--10:15 AM
Luigi Ambrosio
Some recent progress on the theory of metric measure spaces with ,
lower bounds on Ricci curvature
10:15--10:45 AM
Coffee-Tea Break
10:45--12:15 AM
Takis Souganidis
Stochastic homogenization of Hamilton Jacobi and Bellman equations
Free afternoon
Day 4-June 7th, 2012, Thursday
Chair
Takis Souganidis
9:00--10:30 AM
Henri Berestycki
Traveling waves guided by the medium in models of medicine and
biology
10:30--11:00 AM
Coffee-Tea Break
11:00--11:45 AM
Changyou Wang
$C^1$-boundary regularity of planar infinity harmonic functions
LUNCH
Chair
Guozhen Lu
2:00--2:45 PM
Chun Liu
Energetic variational approaches: onsager's maximum dissipation
principle, general diffusion, optimal transport and stochastic integrals.
2:45--3:15 PM
Coffee-Tea Break
3:15--4:00 PM
Xiaoping Yang
On regularity of minimizing geodesics in sub-riemannian manifolds
4:00--4:15 PM
Short break
4:15--5:00 PM
Enrico Valdinoci
Nonlocal nonlinear variational problems
6
Day 5-June 8th, 2012, Friday
Chair
Fanghua Lin
8:45--9:30 AM
Jun-cheng Wei
Stable and unstable solutions of Allen-Cahn equation
9:30--10:00 AM
Coffee-Tea Break
10:00--10:45 AM
Juan Luis Vazquez
Nonlinear diffusion equations involving fractional Laplacians
10:45--11:00 AM
Short break
11:00--11:45 AM
Chongchun Zeng
Unstable manifolds and $L^2$ nonlinear instability of Euler equations
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ABSTRACT
90-minutes Speakers:
Luigi Ambrosio
E.N.S. Pisa, Italy
E-mail: [email protected]
Title: Some recent progress on the theory of metric measure spaces with , lower bounds on
Ricci curvature
Abstract:
In the lecture I will illustrate recent progress on the theory of metric measure spaces with
curvature bounded from below, initiated by Sturm and Lott & Villani. In particular I will focus
on the power of optimal transportation theory, leading to calculus tools and Sobolev space
theory in metric measure spaces. As an application, I will present the theory of RCD spaces
introduced in recent joint work with Gigli and Savarè. If time permits, connections with
Gamma-calculus and with the theory of Dirichlet forms will also be illustrated.
Henri Berestycki
University of Chicago, USA
E-mail: [email protected]
Title: Traveling waves guided by the medium in models of medicine and biology
Abstract:
This lecture is about fronts and propagation phenomena for some classes of reaction-diffusion
equations in non-homogeneous media. I will discuss three models arising in population
dynamics and in medicine in which the medium imposes a direction of propagation.
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Luis Caffarelli
University of Texas, Austin, USA
E-mail: [email protected]
Title: A counter-example in 3D and homogenization of mean curvature motions in 2D
Abstract: We will discuss a counter-example to the homogenization of the forced mean
curvature motion in a periodic setting in dimension N ≥ 3 when the forcing is positive and a
general homogenization result of geometric motions in dimension N = 2 under the assumption
that there exists a constant δ > 0 such that every straight line moving with a normal velocity
equal to δ is a subsolution for the motion. This is joint work with regis Monneau.
Chang-Shou Lin
National Taiwan University, Taiwan
E-mail: [email protected]
Title: Elliptic integrable system: A case study
Abstract:
In principal, solutions of integrable system can be expressed via holomorphic data, at least
locally. This holomorphic data is the so- called developing maps. The monodromic
transformation for developing maps contains important information either from geometric or
algebraic point of view. In some situation when bubbling phenomenon is concerned. The
geometry from the equations governing the locations of blowup points is also closely connected
with the developing maps. In this talk, I will use mean field equations ( Liouville type) as an
example to explain these connections.
Takis Souganidis
University of Chicago, USA
E-mail: [email protected]
Title: Stochastic homogenization of Hamilton Jacobi and Bellman equations
Abstract:
I will present a survey of the theory of homogenization of Hamilton Jacobi and Bellman
equations in random environments. I will also discuss recent results, some applications as well as
rates of convergence.
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45-minutes Speakers:
Xavier Cabre
ICREA and UPC, Barcelona, Spain
E-mail: [email protected]
Title: Elliptic problems with axial symmetry: antisymmetry and isoperimetry
Abstract:
The understanding of some nonlinear elliptic problems for the Laplacian in Rn involves
solutions in Rn = Rm ×Rk which are radially symmetric with respect to the first m variables and
also with respect to the last k variables. Under such axial symmetry, the Laplacian in Rn
becomes an operator in R2 with the homogeneous weight jx1jm�1jx2jk�1. A firrst example is
the Allen-Cahn equation in R2m, for which the existence of an axially symmetric minimizer in
R8 = R4× R4 is still an important open question related to the conjecture of De Giorgi. We will
explain some new results on antisymmetry or oddness of solutions motivated by this open
question. A second example is the understanding of the \explosion problem" in bounded domains
of Rn. Here, the boundedness of stable solutions is expected for n≤9, but this is still an open
problem. The study of this question in domains with axial symmetry has led us to several new
isoperimetric and Sobolev inequalities (with best constants) which involve homogeneous
weights in Rn. Surprisingly, even if our weights are nonradial, balls are still the optimal sets.
Irene Gamba
University of Texas, Austin, USA
E-mail: [email protected]
Title: Solutions to the Boltzmann transport equation for soft potentials
Abstract:
In this talk, we will discuss the well-posedness of solutions to the non-linear Boltzmann
equation with soft potentials for a large class on initial data that is pointwise bounded above and
below by a class of global Maxwellians. When these solutions are global in time, we obtain
results on their large-time asymptotics. We also show the $L^p$ regularity in physical and
velocity space for a given range of values of $p$ depending on the initial data as well as on the
integrability of the angular part of the collision kernel and the strength of the soft potentials.
This is work in collaboration with R.Alonso, C.Bardos and D.Levermore.
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Yanyan Li
Rutgers University, USA
E-mail: [email protected]
Title: Some analytic aspects of conformally invariant fully nonlinear equations
Abstract:
We will discuss some work on conformally invariant elliptic and degenerate elliptic equations
arising from conformal geometry. These include results on Liouville type theorems, Harnack
inequalities, and Bocher type theorems.
Chun Liu
Penn State University, USA
E-mail: [email protected]
Title: Energetic Variational Approaches: Onsager's Maximum Dissipation Principle, General
Diffusion, Optimal Transport and Stochastic Integrals.
Abstract:
In the talk, I will explore the general framework of energetic variational approaches, especially
Onsager's Maximum Dissipation Principles, and their particular in generalized diffusion. We
will discuss the roles of different stochastic integrals (Ito's form, Stratonovich's form and other
possible forms), and the procedure of optimal transport in the context of general framework of
theories of linear responses.
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Yuan Lou
Ohio-State University, USA
E-mail: [email protected]
Title: Evolution of dispersal in heterogeneous landscapes
Abstract:
A general question in the study of the evolution of dispersal is what kind of dispersal strategies
can convey competitive advantages and thus will evolve. We will discuss some recent works on
reaction-diffusion-advection model for two competing species, in which the species are assumed
to have the same population dynamics but different dispersal strategies. We found a conditional
dispersal strategy which results in the ideal free distribution of species, and we investigate
whether such dispersal strategy is evolutionary stable. Discrete and nonlocal dispersal models
will also be discussed if time allows.
Guozhen Lu
Wayne State University, USA
E-mail: [email protected]
Title: A new approach to sharp Moser-Trudinger and Adams inequalities in unbounded domains:
A rearrangement-free method
Abstract: We will discuss some recent results in the direction of sharp constants for the
Moser-Trudinger inequality and Adams inequality in high order Sobolev spaces. We develop a
new method to prove such inequalities in unbounded domains without using symmetrization.
This argument is substantially different from those in the literature where symmetrization was
crucial. It enables us to derive Adams inequalities in high order Sobolev spaces of arbitrary
orders and thus prove such theorems in full generality. It can also be used to obtain sharp
Moser-Trudinger inequalities in other more general settings including the Heisenberg group
where symmetrization is not available. This is joint work with Nguyen Lam.
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Luis Silvestre
University of Chicago, USA
E-mail: [email protected]
Title: On the continuity (or discontinuity) of solutions of drift-diffusion equations
Abstract:
We study parabolic equations which consist on a drift term (a vector field times the gradient)
plus a diffusion term (either the Laplacian or the fractional Laplacian). We analyze what
assumptions on the drift would assure that the solution remains continuous for positive time. A
particularly interesting case is when the drift is a divergence free vector field, since these appear
frequently in equations from fluid mechanics. Assuming that the drift is bounded respect to some
norm which is invariant by the scaling of the equation gives Holder continuity estimates in many
cases. We will prove that when this scaling condition is violated, discontinuities can form in
finite time, even if the drift is divergence free. A notable exception is for an equation with
classical diffusion (with the usual Laplacian), in 2 space dimensions, and a drift which is
independent of time. For that case we obtain a modulus of continuity for any divergence free
drift in $L^1$ (which is highly supercritical).
Zhongwei Shen
University of Kentucky, USA
E-mail: [email protected]
Title: Convergence Rates in Periodic Homogenization
Abstract:
In this talk I will describe some of recent progress on the rates of convergence of solutions in
L^p and W^{1,p} in periodic homogenization. Both Dirichlet and Neumann boundary conditions
are considered. The asymptotic expansions of Green and Neumann functions as well as
convergence rates of eigenvalues will also be discussed. This is a joint work with Carlos Kenig
and Fanghua Lin.
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Juan Luis Vazquez
Universidad Autonoma De Madrid
E-mail: [email protected]
Title: Nonlinear diffusion equations involving fractional Laplacians
Abstract:
Recent research in the area of nonlinear diffusion has focused on nonlinear elliptic and
parabolic equations involving fractional Laplacian operators or other similar nonlocal operators.
We present a model for flow in porous media including nonlocal (long-range) diffusion effects
of such type. The first one is based on Darcys law and the pressure is related to the density by
an inverse fractional Laplacian operator. It is simplest form it reads ut =∇(u∇(−Δ)�su) with 0
< s < 1. We prove existence of solutions that propagate with finite speed, which is unexpected
in fractional diffusion models. The model has also the very interesting property that mass
preserving self-similar solutions can be found by solving an elliptic obstacle problem, and they
are asymptotic attractors of the flow. We also prove C_ regularity for 0 < s < 1, that is lost in
the limit s = 1. We will finally comment on some alternative models for nonlocal, nonlinear
diffusion.
Enrico Valdinoci
Universita' Di Milano, Italy
E-mail: [email protected]
Title: Nonlocal nonlinear variational problems
Abstract:
I would like to discuss some questions related to some semilinear equations driven by a
nonlocal elliptic operator (for example, the Allen-Cahn equation, in which the classical Laplace
operator is replaced by a fractional Laplacian). In particular, I would like to study the
qualitative properties of the solutions, such as symmetry, density estimates of the level set,
asymptotic behaviors, etc.
The limit interfaces of these problems are related to both the local and the nonlocal perimeter
functionals: on this topic, I would like to discuss some rigidity and regularity results and to
present some open problems.
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Lihe Wang
University of Iowa/Shanghai Jiao Tong University
E-mail: [email protected]
Title: Boundary regularity for quasiconvex functional
Abstract:
We will talk about the H\"older regularity for a class of operators. This is a joint work
with S. Byun and K. Hun.
Changyou Wang
University of Kentucky, USA
E-mail: [email protected]
Title: $C^1$-boundary regularity of planar infinity harmonic functions
Abstract:
In this talk, I will discuss the boundary regularity issue for infinity harmonic functions.This
includes (i) the boundary differentiability in all dimensions; and (ii) the $C^1$-boundary
regularity in dimension two. The former extends the interior differentiability theorem by
Evans-Smart; and the latter extends the previous work by Savin, and Evans-Savin. This is a joint
work with Yifeng Yu.
Jun-cheng Wei
Chinese University Hong Kong, Hong Kong
E-mail: [email protected]
Title: Stable and unstable solutions of Allen-Cahn equation
Abstract:
Given a minimizing cone $C$ in $R^n$ with negative indicial root, we prove that there exist
bounded solutions of Allen-Cahn equation whose zero sets are asymptotic to $C$ at infinity. As a
consequence, for $ n \geq 8$, there exists stable solutions of Allen-Cahn equation whose level sets
are not hyperplanes. Then we discuss various unstable solutions in $R^2$, including solutitions
whose zero sets consist of arbitrarily many intersecting lines. (Joint work with Frank Pacard, and
Liu Yong.)
15
Xiao-Ping Yang
Nanjing University of Science & Technology, China
E-mail: [email protected]
Title: On Regularity of Minimizing Geodesics in Sub-Riemannian Manifolds
Abstract:
In this talk, We will discuss geodesics in sub-Riemannian manifolds. In Carnot groups of step_ 3,
all sub-Riemannian geodesics are proved to be normal. The proof is based on a reduction argument
and the Goh condition for minimality of singular curves. The smoothness of abnormal minimizers
of sub-Riemannian manifolds of step 3 with a nilpotent basis is shown. This is a joint work with
Kanghai Tan.
Chongchun Zeng
Georgia Tech, USA
E-mail:[email protected]
Title: Unstable manifolds and $L^2$ nonlinear instability of Euler equations
Abstract:
We consider the nonlinear instability of a steady state $v_{0}$ of the Euler equation in an $n$-dim
fixed bounded domain. When considered in $H^s$, $s>1$, at the linear level, the stretching of the
steady fluid trajectories induces unstable essential spectrum which corresponds to linear instability
at small spatial scales and the corresponding growth rate depends on the choice of the space
$H^s$. Therefore, more physically interesting linear instability relies on the unstable eigenvalues
which correspond to large spatial scales. In the case when the linearized Euler equation at
$v_0$ has an exponential dichotomy of center-stable and unstable (from eigenvalues) directions,
most of the previous results obtaining the expectednonlinear instability in $L^2$ (the energy
space) were based on the vorticity formulation and therefore only work in 2-dim. In this talk, we
prove, in any dimensions, the existence of the unique local unstable manifold of $v_0$, under
certain conditions, and thus its nonlinear instability. Our approach is based on the observation that
the Euler equation on a fixed domain is an ODE on an infinite dimensional manifold of volume
preserving maps in in function spaces. This is a joint work with Zhiwu Lin.
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INVITED SPEAKERS
90-minutes Speakers:
Name
Luigi Ambrosio
Henri Berestycki
Affiliation
E.N.S. Pisa, Italy
University of Chicago
E-mail
[email protected]
[email protected]
Luis Caffarelli
Chang-Shou Lin
Takis Souganidis
University of Texas, Austin, USA
National Taiwan University, Taiwan
University of Chicago, USA
[email protected]
[email protected]
[email protected]
45-minutes Speakers:
Name
Xavier Cabre
Irene Gamba
Yanyan Li
Chun Liu
Yuan Lou
Guozhen Lu
Luis Silvestre
Zhongwei Shen
Juan Luis Vazquez
Enrico Valdinoci
Lihe Wang
Changyou Wang
Jun-cheng Wei
Xiao-Ping Yang
Chongchun Zeng
Affiliation
ICREA and UPC, Barcelona
University of Texas, Austin, USA
Rutgers University, USA
Penn State University, USA
Ohio-State University, USA
Wayne State University, USA
University of Chicago, USA
University of Kentucky, USA
Universidad Autonoma De Madrid
Universita' Di Milano
University of Iowa, USA
University of Kentucky, USA
Chinese University Hong Kong
Nanjing University of Science
& Technology, China
Georgia Tech, USA
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E-mail
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
GNERAL INFORMATION
LOCATION
Lecture hall, on Fourth Floor
Sir Run Run Shaw Business Administration,
Yuquan Campus, Zhejiang University
BANQUET
Date: June 5, 2012
Time: 6:00 PM
Venue: Shanwaishan Restaurant
Date: June 6, 2012
Time: 6:00 PM
Venue: Weizhuang @ Lingyin Restaurant
BREAKFAST
Date: June 4-8, 2012
Time: 8:00 AM
Venue: Jinxi Hotel
LUNCH
Date: June 4-8, 2012
Time: 12:00
Venue: Shaw Science Building Restaurant
DINNER
Date: June 4, 7, 8, 2012
Time: 6:00 PM
Venue: Jinxi Hotel
HOTEL
Jinxi Hotel
Address: No. 39 Causeway Yang, Hanghzou
Tel: (86) 0571-87992288-6901
Homepage: http://www.jinxihotel.com/weben/Public/Default.aspx
PICK UP (Jinxi Hotel to Yuquan Campus, Zhejiang University)
Jinxi Hotel Main Gate
Time: 8:20 AM, June 4-8, 2012
CONTACT:
Mr. Liping Zhou
Tel: 0571-87953867(Office)
Mobile: 13336187298
E-mail: [email protected]
Miss Sujing Li
Tel: 0571-87953947
Mobile: 15869192195
E-mail: [email protected]
CONFERENCE WEBSITE
http://www.math.zju.edu.cn/ICACM/PNPDE/
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Map of Yuqan Campus, Zhejiang University
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