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SCHEDULE
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| 8:00 - 9:15 |
REGISTRATION & CONTINENTAL BREAKFAST
*LAST DAY TO PURCHASE BANQUET TICKETS* |
| 9:15 |
WELCOME |
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9:15 - 10:00
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Y. Brenier (CNRS, Laboratoire Dieudonne)
Asymptotic Analysis of the Born-Infeld Electromagnetism
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10:05 - 10:50
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G. Alberti (Pisa)
Microsctructures in a Model of Di-block Copolymers Melt |
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10:50 - 11:20
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MORNING COFFEE |
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11:20 - 12:05
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Y. Grabovsky (Temple)
A Generalized Theorem of Chandler Davis |
| 12:05 - 12:30 |
Short Talk - H. Jiang (New
York)
Remarks on a Singular Elliptic Equations |
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12:30 - 2:30
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LUNCH |
| 2:30 - 3:15 |
P. Bauman (Purdue)
Variational Methods for Analyzing Phase Transitions in Chiral
Liquid Crystals |
| 3:20 - 4:05 |
P. Sternberg (Indiana)
Stable Vortex Solutions to the Ginzburg-Landau Energy |
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4:05 - 4:30
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AFTERNOON TEA |
| 4:30 - 4:55 |
Short Talk - T. Giorgi
(New Mexico State)
Superconductors Surrounded by Normal Materials |
| 4:55 - 5:20 |
Short Talk - J. A. Montero (McMaster)
Stable vortex solutions to the Ginzburg Landau Energy |
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5:20 - 5:45
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Short Talk - X. Kang (Toronto)
Localization Properties for a Porous Medium Equation with Source
Term |
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5:45 - 7:30
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WELCOME RECEPTION |
| 9:00 - 9:15 |
CONTINENTAL BREAKFAST |
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9:15-10:00
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F. Otto (Bonn)
Multiscale Analysis in Micromagnetism |
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10:05 -10:50
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C. De Lellis
(MPI Leipzig)
Nonlinear Versions of the BV Structure Theorem and of Vol'pert
Chain
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10:50 -11:20
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MORNING COFFEE |
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11:20 -12:05
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E. Sandier (Paris-12)
Asymptotics of the Time Dependant Ginzburg-Landau Equations |
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12:05 - 12:30
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Short Talk - N. Ahmad (Toronto)
Geometry of Shape Recognition Via Optimal Transportation
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12:30 - 2:30
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LUNCH |
| 2:30 - 3:15 |
D. Smets (Paris-VI)
Mean Curvature Flows and the Parabolic Ginzburg-Landau Equation |
| 3:20 - 4:05 |
A. Aftalion (Paris-VI)
Properties of Vortices in Rotating Bose Einstein Condensates
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4:05 - 4:30
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AFTERNOON TEA |
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4:30 - 4:55
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Short Talk - M. Moakher (National
Engineering School at Tunis)
Rods with Microstructure as a Model for Double-Stranded Rods
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| 6:30 - |
BANQUET AT GOLDFISH RESTAURANT |
| 9:00 - 9:15 |
CONTINENTAL BREAKFAST |
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9:15 - 10:00
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N. Ghoussoub (British
Columbia)
A Variational Principle for Dissipative Evolution Equations |
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10:05 - 10:50
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G. Tarantello (Roma-Tor Vergata)
Liouville-type Equations in Gauge Field Theory
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10:50 - 11:20
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MORNING COFFEE |
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11:20 - 12:05
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D. Kinderlehrer
(Carnegie Mellon)
The Mesoscale View of Grain Growth |
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12:05 - 12:30
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Short Talk - I. Blank
(Rutgers)
Eliminating Mixed Asymptotics in Obstacle Type Free Boundary
Problems |
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Afternoon free |
| 9:00 - 9:15 |
CONTINENTAL BREAKFAST |
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9:15 - 10:00
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S. Serfaty (Courant
Institute)
Asymptotics of the Time Dependant Ginzburg-Landau Equation |
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10:05 - 10:50
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I. Shafrir (Technion)
The Logarithmic HLS Inequality for Systems on Compact Manifolds
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10:50 - 11:20
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MORNING COFFEE |
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11:20 - 12:05
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M. Kowalczyk (Kent State) |
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12:05 - 12:30
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Short Talk - G. Menon
(Wisconsin)
Dynamic Scaling in Smoluchowski's Coagulation Equation |
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12:30 - 2:30
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LUNCH |
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2:30 - 3:15
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F. Bethuel (Paris-VI)
A Survey on Some New Results for Travelling Waves of the Gross-Pitaevskii
Equation |
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3:20 - 4:05
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D. Spirn (Brown)
Dynamics and Instability of Elliptical Vortex Patches |
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4:05 - 4:30
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AFTERNOON TEA |
| 4:30 - 4:55 |
Short Talk - F. Ting (Toronto)
Stability of Pinned Vortices of the Ginzburg Landau Equations
with External Potential |
| 4:55 - 5:20 |
Short Talk - C. Lin (National
Cheng Kung)
Homogenization of the Dirac System |
|
5:20 - 5:45
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Short Talk - L. Novozhilova
(MSU)
Global Injectivity and Partial Regularity of Axisymmetric Minimizers
in Nonlinear Elasticity |
| 5:45 - 6:10 |
Short Talk - H. Jadallah
(Purdue)
TBA |
| 9:00 - 9:15 |
CONTINENTAL BREAKFAST |
|
9:15 - 10:00
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S. Gustafson (British
Columbia)
On the Dynamics of Vortices and Solitary Waves |
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10:05 - 10:50
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G. Dolzmann (Maryland)
Nonconvex Variational Problems and Minimizing Young Measures
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10:50 - 11:20
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MORNING COFFEE |
| 11:20 - 11:45 |
G. Auchmuty (Houston)
Variational Principles for Non-potential Problems |
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11:50 - 12:35
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G. Friesecke (Warwick)
Variational Methods in Quantum Chemistry
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afternoon free |
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ABSTRACTS
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Aftalion, Amandine
(Universite Paris-VI)
Properties of Vortices in Rotating Bose Einstein Condensates
We consider a rotating Bose-Einstein condensate in a harmonic
trap and investigate the behavior of the wave function which solves
the Gross Pitaevskii equation. We give a simplified expression
of the Gross-Pitaevskii energy in an asymptotic regime, which
only depends on the number and shape of the vortex lines. Following
recent experiments, we study in detail the line of a single quantized
vortex, which has either a $U$ or $S$ shape.
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Ahmad, Najma (University
of Toronto)
Geometry of Shape Recognition via Optimal Transportation
A Monge-Kantorovich optimal transportation problem between measures
supported on the boundaries of domains in ${\mathbb R}^2$ is studied
with the intent to get an insight into the underlying geometry of
a shape recognition problem in computer vision --- where one wants
to match two simple closed planar curves. The focus is on investigating
(i) uniqueness, (ii) smoothness and (iii) geometrical characterization
of the solutions. Optimality of these solutions is measured against
a cost function defined between the two curves to be compared. Topological
constraints allow (iv) a classification of the cost function that
strongly dictates the geometry of the optimal solutions.
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Alberti, Giovanni (Universita
di Pisa)
Microsctructures in a Model of Di-block Copolymers Melt
I will describe some mathematical features of a variational
model for the description of micro-phase separation in di-block
copolymer melts. In dimension one, minimizers of this energy functional
present periodic patterns on a certain microscopic scale. In higher
dimension, however, very little is known on the structure of minimizers.
In a joint work with R. Choksi and F. Otto we have proved a uniform
energy bound (on the right microscopic scale), and shown that the
admissible patterns should arise as local minimizer on the entire
space of the unscaled functional.
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Auchmuty, Giles (Houston)
Variational Principles for Non-potential Problems
There is a large class of variational principles based on the Young-Fenchel
inequality for dual convex functionals. These variational principles
are different in many ways to classical variational principles.
In this talk the general form of these principles and three specific
examples will be described. The examples are linear non-self adjoint
equations which satisfy a Lax-Milgram property, the Brezis-Ekeland
variational principle for the heat equation and the variational
formulation of finite dimensional variational inequalities.
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Bauman, Patricia (Purdue
University)
Variational Methods for Analyzing Phase Transitions in Chiral
Liquid Crystals
We introduce the Landau-de Gennes free energy used to model the
transition between chiral nematic and smectic A liquid crystal phases.
Within this mathematical framework, the physically observed growth
behavior of the twist and bend Frank constants in the energy play
a major role in bringing about the transition. We rigorously establish
a transition regime separating the two phases, using variational
techniques to analyze two competing effects: the layer formation
of the smectic phase and the twist tendency of the chiral nematic
phase. Our discussion will illustrate the analogies as well as the
discrepancies in modeling and behavior between smectic A liquid
crystals and superconducting materials described by the Ginzburg-Landau
theory.
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Bethuel, Fabrice (Universite
Paris VI)
A Survey on Some New Results for Travelling Waves of the Gross-Pitaevskii
Equation
We present some joint work with G. Orlandi and D. Smets as well
as some new results by P.Gravejat concerning the existence problem
and qualitative propoerties of travelling waves of the GP equation.
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Blank, Ivan (Rutgers University)
Eliminating Mixed Asymptotics in Obstacle Type Free Boundary
Problems
We show a method to eliminate a type of mixed asymptotics in certain
free boundary problems and give two examples of its application.
It appears that these problems cannot be handled by the monotonicity
formula of Alt, Caffarelli, and Friedman (1984), or by the more
recent monotonicity formula of Caffarelli, Jerison, and Kenig (2002).
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Brenier, Yann (CNRS,
Laboratoire Dieudonne)
Asymptotic Analysis of the Born-Infeld Electromagnetism
Born and Infeld introduced in 1934 a non linear version of the Maxwell
equations, which is still for use in high energy physics. Remarkably
enough, the Born-Infeld system can be enlarged as a 10x10 system
of hyperbolic conservation laws, quite similar to the classical
MHD equations, with a nearly quadratic conserved energy. This allows
us to perform some asymptotic analysis by using a relative entropy
method going back to Dafermos.
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DeLellis, Camillo (Max-Planck-Institute)
Nonlinear Versions of the BV Structure Theorem and of Vol'pert
Chain
In the last fifteen years some physical models have raised the issue
of understanding the singular limit of certain families of smooth
functionals which involve first and second derivatives. It turned
out that these problems lead naturally to the study of (nonsmooth)
divergence free $m:{\bf R}^2 \to {\bf S}^1$ such that ${\rm div}\,
\Phi (m)$ is a Radon measure for any $\Phi$ belonging to appropriate
classes of vector fields. When $m$ is a function of bounded variation,
${\rm div}\, \Phi (m)$ can be computed by using Vol'pert chain rule.
Though general $m$'s are far (in terms of linear function spaces)
from having bounded variation, in a joint work with Felix Otto we
have shown that the pointwise behavior of $m$ is similar to that
of BV functions. Hence it would be natural to expect that ${\rm
div}\, \Phi (m)$ can be computed in a similar fashion. It turns
out that very similar questions arise naturally in different areas
of PDE's. We will give a brief overview and we will show recent
results giving affirmative answers to some of them.
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Dolzmann, Georg (University
of Maryland)
Nonconvex Variational Problems and Minimizing Young Measures
Variational integrals modeling solid-to-solid phase transformations
often fail to be weakly lower semicontinuous because the energy
densities $f$ are not quasiconvex in the sense of Morrey. In this
talk we analyse properties of minimizing Young measures generated
by minimizing sequences for these variational integrals. We prove
that the moments of order $q>p$ exist if the integrand is sufficiently
close to the $p$-Dirichlet energy at infinity. A counterexample
related to the one-well problem in two dimensions shows that one
cannot expect in general $L^\infty$ estimates, i.e., that the support
of the minimizing Young measure is uniformly bounded.
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Friesecke, Gero (University
of Warwick)
Variational Methods in Quantum Chemistry
Recent successes of variational methods in quantum chemistry include
(i) a new and simpler proof of Zhislin's fundamental structure theorem
on the spectrum of many-particle Schr"odinger operators,(ii)
a generalization of Zhislin's result to a central approximate method
of quantum chemistry (the multiconfiguration self-consistent field
method, which may be viewed as a closure assumption on higher oder
correlations in terms of lower order correlations), (iii) a rigorous
derivation of the celebrated van der Waals 1/r^6 law for long range
interatomic forces from the many-electron Schr"odinger equation.
The last result is joint work with Phil Gardner (Warwick).
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Ghoussoub, Nassif (University
of British Columbia & The Pacific Institute for the Mathematical
Sciences)
A Variational Principle for Dissipative Evolution Equations
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Giorgi, Tiziana (New Mexico
State University)
Superconductors Surrounded By Normal Materials
We study questions related to existence in suitable weighted Sobolev
spaces, and to properties of minimezers of a generalized Ginzburg-Landau
energy functional, which models a bounded superconductor surrounded
by a normal material. The model in consideration is of interest
as the effects of superconducting electron pairs diffusing into
the normal region are here represented.
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Grabovsky, Yury (Temple
University)
A Generalized Theorem of Chandler Davis
A polycrystal is a mixture of anisotropic materials (crystals) where
each material may participate in a composite in any orientation.
The effective conductivity tensor of such a composite depends on
the microstructure of the composite. The set of effective properties
one can obtain by mixing the same set of materials in different
ways is called the G-closure of the original materials. The G-closure
set has two important qualities: SO(3) invariance and a certain
convexity property. In order to understand the interplay between
these two properties we would like to understand SO(3) invariant
functions with the convexity property. The first such result is
due to Chandler Davis. In our case we examine what happens when
the group action in Davis's theorem is non-linear. In the process
we uncover a simple abstract mechanism behind the Davis's classical
theorem. Our generalization features arbitrary groups, non-linear
group actions and infinite dimensional vector spaces. We also gain
extra flexibility to prove convexity of some G-invariant convex
functions even though the theorem does not hold for all such functions.
Even in the case of linear group actions on finite dimensional spaces
we achieve a new generalization of Davis's result.
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Gustafson, Stephen
(University of British Columbia)
On the Dynamics of Vortices and Solitary Waves
We present results describing the dynamics of stable, localized
structures in solutions of nonlinear evolutions PDEs. The main examples
are superconducting vortices (Ginzburg-Landau equations) and solitary
waves (nonlinear Schroedinger equations).
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Jadallah, Hala (Purdue
University)
TBA
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Jiang, Huiqiang (New York
University)
Remarks on Singular Elliptic Equations
We consider nonnegative solutions of a singular elliptic equation,
which arises in thin film rupture and minimal surface theory. We
get a general estimate of the size of singular (zero) set.
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Kang, Xiaosong (University
of Toronto)
Localization Properties for a Porous Medium Equation with Source
Term
We establish the strict localization of a porous medium equation
with source, i.e., if the initial data is compactly supported, the
unbounded solution will be of uniformly compact support. Our argument
works for arbitrary spatial dimension, hence the result extends
the well-known one dimensional case.
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Kinderlehrer, David
(Carnegie Mellon University)
The Mesoscale View of Grain Growth
Most technologically useful materials are polycrystalline, composed
of many small crystallites called grains separated by interfaces
called grain boundaries. These grain boundaries play a role in many
material properties, for example conductivity and fracture toughness,
and across many scales. Preparing arrangements or distributions
of boundaries suitable for a given purpose is a central problem
in materials. It is, indeed, the central problem of microstructure
and has an extensive history dating from prehistory. Grain growth
is one of the primary microstructural mechanisms. We may ask many
questions, for example, to what extent is grain growth like or unlike
the growth of soap bubbles. We discuss some of the scientific challenges
we encounter in the investigation of these issues. In recent years
we have been able to begin simulations at mesoscale which are both
accurate and statistically significant, that is, they are very large
scale. What is the 'answer' of such a simulation? This is a very
pregnant question. We present various results and surprises, but
primarily we expose the rich trove of problems this study is unveiling.
This is joint work with Florin Manolache, Jeehyun Lee, Irene Livshits,
Gregory Rohrer, Anthony Rollett, and Shlomo Ta'asan. [1] Partially
supported by the NSF under the MRSEC program.
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Lin, Chi-Kun (National Cheng
Kung University)
Homogenization of the Dirac System
The homogenization of Dirac system is studied. It generates memory
effects. The memory (or nonlocal) kernel is described by the Fredholm
integral equation. When the coefficient is independent of space,
the nonlocal kernel can be characterized explicitly in terms of
Young's measure. The homogenized equation can be reformulated in
the kinetic form by introducing the kinetic variable.
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Menon, Govind (University
of Wisconsin)
Dynamic Scaling in Smoluchowski's Coagulation Equation
Smoluchowki's coagulation equations describe a wide variety of mass
agggregation processes in physical chemistry and physics (polymerization,
colloidal separation, aerosol physics, gravitational clustering...).
They also arise in population genetics and combinatorics. I will
describe simple proofs of optimal results on dynamic scaling in
these equations. These involve one-parameter families of self-similar
solutions with fat tails, and the characterization of their domains
of attraction. This is work with Bob Pego (Maryland).
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Moakher, Maher (National
Engineering School at Tunis)
Rods with Microstructure as a Model for Double-Stranded Rods
I will present a continuum theory for birods composed of two thin
elastic rods, here termed strands, that are bound together by elastic
forces. The birod is modeled as a special Cosserat macro-rod endowed
with microstructure parameters that give the relative positions
and orientations of the strands with respect to the position and
orientation of the macro-rod. Constitutive relations and the equations
of motion for the birod are derived from a variational principle.
Possible applications of this theory to modelling deformation of
the DNA double helix will be discussed.
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Montero, Jose Alberto
(McMaster)
Stable Vortex Solutions to the Ginzburg Landau Energy
Using the theory of weak Jacobians and a gamma-convergence argument
we establish the existence of local minimizers to the Ginzburg Landau
energy with a magnetic field in certain non-convex, simply-connected
domains in 3-D. This is Joint work with Robert Jerrard and Peter
Sternberg.
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Novozhilova, Lidiya (MSU)
Global Injectivity and Partial Regularity of Axisymmetric Minimizers
in Nonlinear Elasticity
Global injectivity of axisymmetric deformations for a class of incompressible
hyperelastic materials is proved under the axisymmetric counterpart
of the injectivity condition by Ciarlet and Ne\cap{c}as. Higher
regularity properties of the radial and axial components are also
established using some results from geometric function theory.
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Otto, Felix (Bonn)
Multiscale Analysis in Micromagnetism
Domains and walls in ferromagnets are a paradigm for pattern formation
in materials science. Domains are subregions of the sample $\Omega$
in which the magnetization $m$ is nearly constant; the transition
layers separating domains are called walls. We will focus on the
technologically important ferromagnetic films. Mathematically speaking,
the micromagnetic model is a non--convex, non--local variational
problem for the magnetization $m$. It is characterized by several
length scales: On one end, there are the scales given by the sample
geometry (film thickness and film diameter) and on the other end,
there are the scales which depend only on the material. This set--up
drives the pattern formation on intermediate scales. In this lecture,
we shall try to explain specific experimental observations on walls
and domains in ferromagnetic films starting from the micromagnetic
model. First, we shall try to understand domain formation neglecting
wall energy. Then, we'll take wall energy into account and will
discover that there are different modes of walls. Finally, we'll
have to take wall interaction into account. We will use a mixture
of heuristic and rigorous arguments and shall present some numerical
simulations.
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Sandier, Etienne (Universite
Paris 12 Val de Marne)
Asymptotics of the Time Dependant Ginzburg-Landau Equations
In a joint work with Sylvia Serfaty we extend previous results on
the asymptotics of parabolic Ginzburg-Landau equations for large
kappa to the case of an applied magnetic field of the order of log(kappa).
This involves a new product estimate useful in both static and time
dependent situations.
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Serfaty, Sylvia (Courant
Institute)
Asymptotics of the Time Dependant Ginzburg-Landau Equations
In a joint work with Etienne Sandier we extend previous results
on the asymptotics of parabolic Ginzburg-Landau equations for large
kappa to the case of an applied magnetic field of the order of log(kappa).
This involves a new product estimate useful in both static and time
dependent situations.
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Shafrir, Itai (Technion
- Israel Institute of Technology)
The Logarithmic HLS Inequality for Systems on Compact Manifolds
Let $\mathcal M$ be a compact $m$-dimensional Riemannian manifold.
Given a $n\times n$ symmetric matrix $A=(a_{i,j})$ with $a_{i,j}\geq
0$, $\forall i,j$, we give optimal conditions on the vector ${\bf
M}=(M_1,\ldots,M_n)\in{\mathbb R}_{+}^n$ which ensure boundedness
from below of the functional $$ \Psi({\boldsymbol\rho})=\sum_{i=1}^n
\int_{\mathcal M} \rho_i\ln\rho_i+\sum_{i,j=1}^n a_{i,j} \int_{\mathcal
M}\!\int_{\mathcal M} \rho_i(x)\ln d(x,y) \rho_j(y)\,dx\,dy $$ over
$$ \boldsymbol\Gamma_{{\bf M}}=\big\{(\rho_1,\ldots,\rho_n)\in ({\mathcal{L}\ln\mathcal{L}}(\mathcal{M},\mathbb{R}_+))^n,\,\int_{\mathcal{M}}\rho_i=M_i,\,\forall
i\big\}. $$ This result generalizes the logarithmic Hardy-Littlewood-Sobolev
inequality of Beckner to the systems case. In some cases we also
address the question of existence of minimizers. This is a joint
work with Gershon Wolansky.
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Smets, Didier (Universite
de Paris VI)
Mean Curvature Flows and the Parabolic Ginzburg-Landau Equation
We will discuss some issues concerning the evolution of the limiting
defect measures associated to the parabolic Ginzburg-Landau equation
in the whole space.
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Spirn, Daniel (Brown University)
Dynamics and Instability of Elliptical Vortex Patches
We describe the dynamics of elliptical vortex patch by formulating
a nonlinear equation for the boundary of a perturbed patch. In the
regime for which the linearized equation of motion is unstable,
the nonlinear dynamics of a large class of initial perturbations
are determined by the fastest growing mode for the corresponding
linearized equation. In particular, we show that elliptical patches
are unstable in the full nonlinear sense.
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Sternberg, Peter (Indiana
University)
Stable Vortex Solutions to the Ginzburg-Landau Energy
We establish the existence of locally minimizing vortex solutions
to the reduced and full Ginzburg-Landau energy in three dimensional
simply-connected domains with or without the presence of an applied
magnetic field. The approach is based upon the theory of weak Jacobians
and applies to nonconvex sample geometries for which there exists
a configuration of locally shortest line segments with endpoints
on the boundary. This is joint work with Robert Jerrard, Alberto
Montero and William Ziemer.
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Tarantello, Gabriella
(Roma-Tor Vergata Universita)
Liouville-Type Equations in Gauge Field Theory
We shall discuss the role of Liouville-type equations (and systems)
arising in the study of vortices in various gauge firld theories
(e.g. Chern Simons theory, Electroweak theory etc). For this class
of equations, we present concentration-compactness principles and
mass "quantization" properties for the concentration phenomenon
that yield to useful existence results, but also to some interesting
open problems.
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Ting, Fridolin (University
of Toronto)
Stability of Pinned Vortices of the Ginzburg Landau Equations
with External Potential
We study the stability of vortex solutions to the Ginzburg-Landau
equations with external potential in two space dimensions. For smooth
and sufficiently small external potentials, there exists a perturbed
vortex solution centered near the critical point of the potential.
We show that these perturbed vortex solutions (pinned vortices)
are orbitally stable.
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