Difference between revisions of "PDE Geometric Analysis seminar"

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(Replacing page with '= PDE and Geometric Analysis Seminar = The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise. == Seminar Schedul...')
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The seminar will be held  in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
 
The seminar will be held  in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
 
+
== Seminar Schedule Fall 2011 ==
== Seminar Schedule Spring 2011 ==
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
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!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Jan 24
+
|Oct 24
|Bing Wang (Princeton)
+
|Valentin Ovsienko ()
|[[#Bing Wang (Princeton)|
+
|[[#Valentin Ovsienko ()|
  ''The Kaehler Ricci flow on Fano manifold '']]
+
  ''TBA'']]
|Viaclovsky
+
|Marí Beffa
|-
 
|Mar 15 (TUESDAY) at 4pm in B139 (joint wit Analysis)
 
|Francois Hamel (Marseille)
 
|[[#Francois Hamel (Marseille)|
 
''Optimization of eigenvalues of non-symmetric elliptic operators'']]
 
|Zlatos
 
|-
 
|Mar 28
 
|Juraj Foldes (Vanderbilt)
 
|[[#Juraj Foldes (Vanderbilt)|
 
''Symmetry properties of parabolic problems and their applications'']]
 
|Zlatos
 
|-
 
|Apr 11
 
|Alexey Cheskidov (UIC)
 
|[[#Alexey Cheskidov (UIC)|
 
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up'']]
 
|Kiselev
 
|-
 
|Date TBA
 
|Mikhail Feldman (UW Madison)
 
|''TBA''
 
|Local speaker
 
|-
 
|Date TBA
 
|Sigurd Angenent (UW Madison)
 
|''TBA''
 
|Local speaker
 
|-
 
|}
 
== Seminar Schedule Fall 2010 ==
 
{| cellpadding="8"
 
!align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|Sept 13
 
|Fausto Ferrari (Bologna)
 
|[[#Fausto Ferrari (Bologna)|
 
''Semilinear PDEs and some symmetry properties of stable solutions'']]
 
|Feldman
 
|-
 
|Sept 27
 
|Arshak Petrosyan (Purdue)
 
|[[#Arshak Petrosyan (Purdue)|
 
''Nonuniqueness in a free boundary problem from combustion'']]
 
|Feldman
 
|-
 
|Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck.  '''Special day, time & room.'''
 
|Changyou Wang (U. of Kentucky)
 
|[[#Changyou Wang (U. of Kentucky)|
 
''Phase transition for higher dimensional wells'']]
 
|Feldman
 
|-
 
|Oct 11
 
|Philippe LeFloch (Paris VI)
 
|[[#Philippe LeFloch (Paris VI)|
 
''Kinetic relations for undercompressive shock waves and propagating phase boundaries'']]
 
|Feldman
 
|-
 
|Oct 29 Friday 2:30pm, Room: B115 Van Vleck.    '''Special day, time & room.'''
 
|[http://www.ima.umn.edu/~imitrea/ Irina Mitrea] (IMA)
 
|[[#Irina Mitrea |
 
''Boundary Value Problems for Higher Order Differential Operators'']]
 
|[https://www.math.wisc.edu/~wimaw/ WiMaW]
 
|-
 
|-
 
|Nov 1
 
|Panagiota Daskalopoulos (Columbia U)
 
|[[#Panagiota Daskalopoulos (Columbia U)|
 
''Ancient solutions to geometric flows'']]
 
|Feldman
 
|-
 
|Nov 8
 
|Maria Gualdani (UT Austin)
 
|[[#Maria Gualdani (UT Austin)|
 
''A nonlinear diffusion model in mean-field games'']]
 
|Feldman
 
|-
 
|Nov 18 Thursday 1:20pm  Room: 901 Van Vleck '''Special day & time.'''
 
|Hiroshi Matano (Tokyo University)
 
|[[#Hiroshi Matano (Tokyo University)|
 
''Traveling waves in a sawtoothed cylinder and their homogenization limit'']]
 
|Angenent & Rabinowitz
 
|-
 
|Nov 29
 
|Ian Tice (Brown University)
 
|[[#Ian Tice (Brown University)|
 
''Global well-posedness and decay for the viscous surface wave
 
problem without surface tension'']]
 
|Feldman
 
|-
 
|Dec. 8 Wed 2:25pm,  Room: 901 Van Vleck. '''Special day, time & room.'''
 
|Hoai Minh Nguyen (NYU-Courant Institute)
 
|[[#Hoai Minh Nguyen (NYU-Courant Institute)|
 
''Cloaking via change of variables for the Helmholtz equation'']]
 
|Feldman
 
 
|-
 
|-
 
|}
 
|}
 
+
==Abstracts==
== Abstracts ==
+
===Valentin Ovsienko ()===
===Fausto Ferrari (Bologna)===
+
To be posted
''Semilinear PDEs and some symmetry properties of stable solutions''
 
 
 
I will deal with stable solutions of semilinear elliptic PDE's
 
and some of their symmetry's properties. Moreover, I will introduce some weighted Poincaré inequalities obtained by combining the notion of stable solution with the definition of weak solution.
 
 
 
===Arshak Petrosyan (Purdue)===
 
''Nonuniqueness in a free boundary problem from combustion''
 
 
 
We consider a parabolic free boundary problem with a fixed gradient condition
 
which serves as a simplified model for the propagation of premixed equidiffusional
 
flames. We give a rigorous justification of an example due to J.L. V ́azquez that
 
the initial data in the form of two circular humps leads to the nonuniqueness of limit
 
solutions if the supports of the humps touch at the time of their maximal expansion.
 
 
 
This is a joint work with Aaron Yip.
 
 
 
 
 
===Changyou Wang (U. of Kentucky)===
 
''Phase transition for higher dimensional wells''
 
 
 
For a potential function <math>F</math> that has two global minimum sets consisting of two compact connected
 
Riemannian submanifolds in <math style="vertical-align=100%" >\mathbb{R}^k</math>, we consider the singular perturbation problem:
 
 
 
Minimizing <math>\int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right)</math> under given Dirichlet boundary data.
 
 
 
I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic,  as  the parameter <math>\epsilon</math>
 
tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and
 
the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary
 
data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.
 
 
 
===Philippe LeFloch (Paris VI)===
 
''Kinetic relations for undercompressive shock waves and propagating phase boundaries''
 
 
 
I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics.  In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves.
 
 
 
 
 
 
 
===Irina Mitrea===
 
''Boundary Value Problems for Higher Order Differential Operators''
 
 
 
As is well known, many phenomena in engineering and mathematical physics
 
can be modeled by means of boundary value problems for a certain elliptic
 
differential operator L in a domain D.
 
 
 
When L is a differential operator of second order a variety of tools
 
are available for dealing with such problems including boundary integral
 
methods,
 
variational methods, harmonic measure techniques, and methods based on
 
classical
 
harmonic analysis. The situation when the differential operator has higher order
 
(as is the case for instance with anisotropic plate bending when one
 
deals with
 
fourth order) stands in sharp contrast with this as only fewer options
 
could be
 
successfully implemented. Alberto Calderon, one of the founders of the
 
modern theory
 
of Singular Integral Operators, has advocated in the seventies the use
 
of layer potentials
 
for the treatment of higher order elliptic boundary value problems.
 
While the
 
layer potential method has proved to be tremendously successful in the
 
treatment
 
of second order problems, this approach is insufficiently developed to deal
 
with the intricacies of the theory of higher order operators. In fact,
 
it is largely
 
absent from the literature dealing with such problems.
 
 
 
In this talk I will discuss recent progress in developing a multiple
 
layer potential
 
approach for the treatment of boundary value problems associated with
 
higher order elliptic differential operators. This is done in a very
 
general class
 
of domains which is in the nature of best possible from the point of
 
view of
 
geometric measure theory.
 
 
 
 
 
===Panagiota Daskalopoulos (Columbia U)===
 
''Ancient solutions to geometric flows''
 
 
 
We will discuss the clasification of ancient solutions to nonlinear geometric flows.
 
It is well known that ancient solutions  appear as blow up limits  at a finite time 
 
singularity of the  flow.
 
Special emphasis will be given to the 2-dimensional Ricci flow.
 
In this case we will show that ancient  compact solution
 
is either the Einstein (trivial)  or one of the King-Rosenau solutions.
 
 
 
===Maria Gualdani (UT Austin)===
 
''A nonlinear diffusion model in mean-field games''
 
 
 
We present an overview of mean-field games theory and show
 
recent results on a free boundary value problem, which models
 
price formation dynamics.
 
In such model, the price is formed through a game among infinite number
 
of agents.
 
Existence and regularity results, as well as linear stability, will be shown.
 
 
 
===Hiroshi Matano (Tokyo University)===
 
''Traveling waves in a sawtoothed cylinder and their homogenization limit''
 
 
 
My talk is concerned with a curvature-dependent motion of plane
 
curves in a two-dimensional cylinder with spatially undulating
 
boundary.  In other words, the boundary has many bumps and we
 
assume that the bumps are aligned in a spatially recurrent manner.
 
 
 
The goal is to study how the average speed of the traveling wave
 
depends on the geometry of the domain boundary.  More specifically,
 
we consider the homogenization problem as the boundary undulation
 
becomes finer and finer, and determine the homogenization limit
 
of the average speed and the limit profile of the traveling waves.
 
Quite surprisingly, this homogenized speed depends only on the
 
maximal opening angles of the domain boundary and no other
 
geometrical features are relevant.
 
 
 
Next we consider the special case where the boundary undulation
 
is quasi-periodic with ''m'' independent frequencies.  We show that
 
the rate of convergence to the homogenization limit depends on
 
this number ''m''.
 
 
 
This is joint work with Bendong Lou and Ken-Ichi Nakamura.
 
 
 
===Ian Tice (Brown University)===
 
''Global well-posedness and decay for the viscous surface wave
 
problem without surface tension''
 
 
 
We study the incompressible, gravity-driven Navier-Stokes
 
equations in three dimensional domains with free upper boundaries and
 
fixed lower boundaries, in both the horizontally periodic and
 
non-periodic settings.  The effect of surface tension is not included.
 
We employ a novel two-tier nonlinear energy method that couples the
 
boundedness of certain high-regularity norms to the algebraic decay of
 
lower-regularity norms.  The algebraic decay allows us to balance the
 
growth of the highest order derivatives of the free surface function,
 
which then allows us to derive a priori estimates for solutions.  We
 
then prove local well-posedness in our energy space, which yields global
 
well-posedness and decay.  The novel LWP theory is established through
 
the study of the linear Stokes problem in moving domains.  This is joint
 
work with Yan Guo.
 
 
 
 
 
===Hoai Minh Nguyen (NYU-Courant Institute)===
 
''Cloaking via change of variables for the Helmholtz equation''
 
 
 
A region of space is cloaked for a class of measurements if observers
 
are not only unaware of its contents, but also unaware of the presence
 
of the cloak using such measurements. One approach to cloaking is the
 
change of variables scheme introduced by Greenleaf, Lassas, and
 
Uhlmann for electrical impedance tomography and by Pendry, Schurig,
 
and Smith for the Maxwell equation.  They used a singular change of
 
variables which blows up a point into the cloaked region. To avoid
 
this singularity, various regularized schemes have been proposed. In
 
this talk I present results related to cloaking via change of
 
variables for the Helmholtz equation using the natural regularized
 
scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the
 
authors used a transformation which blows up a small ball instead of a
 
point into the cloaked region. I will discuss the degree of
 
invisibility for a finite range or the full range of frequencies, and
 
the possibility of achieving perfect cloaking. If time permits, I will
 
mention some results related to the wave equation.
 
 
 
===Bing Wang (Princeton)===
 
''The Kaehler Ricci flow on Fano manifold ''
 
 
 
We show the convergence of the Kaehler Ricci flow on every 2-dimensional Fano manifold which admits big <math>\alpha_{\nu, 1}</math>
 
or <math>\alpha_{\nu, 2}</math> (Tian's invariants).    Our method also works for 2-dimensional Fano orbifolds.
 
Since Tian's invariants can be calculated by algebraic geometry method,  our convergence theorem implies that one can find new Kaehler Einstein metrics
 
on orbifolds by calculating Tian's invariants.
 
An essential part of the proof is to confirm the Hamilton-Tian conjecture in complex dimension 2.
 
 
 
===Francois Hamel (Marseille)===
 
''Optimization of eigenvalues of non-symmetric elliptic operators''
 
 
 
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of <math>R^n</math>. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
 
 
 
===Juraj Foldes (Vanderbilt)===
 
''Symmetry properties of parabolic problems and their applications''
 
 
 
Positive solutions of nonlinear parabolic problems can have a very complex behavior. However, assuming certain symmetry  conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is 'stable'; more specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. As an application, we show new results on convergence of solutions to a single equilibrium.
 
 
 
===Alexey Cheskidov (UIC)===
 
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up''
 
 
 
The problems of blow-up for Navier-Stokes and Euler equations
 
have been extensively studied for decades using different techniques.
 
Motivated by Kolmogorov's theory of turbulence, we present a new unified
 
approach to the blow-up problem for the equations of incompressible
 
fluid motion. In particular, we present a new regularity criterion which
 
is weaker than the Beale-Kato-Majda condition in the inviscid case, and
 
weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case.
 

Revision as of 16:40, 7 September 2011

PDE and Geometric Analysis Seminar

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Seminar Schedule Fall 2011

date speaker title host(s)
Oct 24 Valentin Ovsienko ()
TBA
Marí Beffa

Abstracts

Valentin Ovsienko ()

To be posted