November

Monday 16

16 hrs.

Matteo Rizzi (SISSA, Italy)

Clifford Tori and the singularly perturbed Cahn-Hilliard equation

Abstract

17hrs.

Panayotis Smyrnelis (CMM)

Connecting orbits of the system $u”=\nabla W(u)$

We will give necessary and sufficient conditions for the existence of
bounded minimal solutions of the system $u”=\nabla W(u)$. We will also
prove the existence of heteroclinic, homoclinic and periodic orbits in
analogy with the scalar case. Finally, we will mention new kinds of
connecting orbits that may occur in the vector case.

Thursday 26 – Saturday 28

SOMACHI (Pucón)

http://www.somachi.cl/encuentro2015

Monday 30

15hrs.

Nicolás Carreño (USM)

Insensitizing controls for the Boussinesq system with a reduced number of controls

Abstract

16hrs.

Felipe Barra (DFI, U. de Chile)
Termodinámica de sistemas cuánticos abiertos.
La evolución de sistemas cuánticos abiertos se puede describir, en muchos casos, con la ecuación de Lindblad. En particular la de algunos sistemas que operan como maquinas térmicas o refrigeradores de escala nanoscópica. Sin embargo, para estudiar las propiedades termodinámicas de estos dispositivos es necesario
entender como la interacción con el medio externo genera la descripción de Lindblad en cuestión. En esta charla discutiré dos escenarios frecuentemente usados para describir la interacción de un sistema con su medio ambiente junto con las diferencias (y confusiones)  que introducen en la descripción termodinámica.

October

Wednesday October 14

16hrs.

Søren Fournais (Aarhus University)

Optimal magnetic Sobolev constants in the semiclassical limit

Abstract Soeren

17hrs.

Matteo Cozzi (University of Milan)

Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

Abstract Matteo

Monday October 19

4:00pm

Remy Rodiac  (PUC)

Ginzburg-Landau type problems with prescribed degrees on the boundary

In this talk we will introduce the Ginzburg-Landau equations with the
so-called semi-stiff boundary conditions. It corresponds to
prescribing the modulus of the unknown $u$ on the boundary, together
with its winding number. This is a model for superconductivity which
is intermediate between the full Ginzburg-Landau model with magnetic
field and the simplified Ginzburg-Landau model without magnetic field
but with a Dirichlet boundary data studied by Béthuel-Brézis-Hélein.
Since the winding number is not continuous for the weak convergence in
$H^{1/2}$, the direct method of calculus of variations fails. This is
a problem with lack of compactness and a bubbling phenomenon appears.
We will then give some existence or non existence results for
minimizers of the Ginzburg-Landau energy with prescribed degrees on
the boundary. In order to do this we are also led to study the
Dirichlet energy with the same type of boundary conditions and we make
a link with minimal surfaces in $R^3$.

5:00pm

Yannick Sire (Johns Hopkins University)
Bounds on eigenvalues on riemannian manifolds

I will describe several recent results with N. Nadirashvili where we
construct extremal metrics for eigenvalues on riemannian surfaces.
This involves the study of a Schrodinger operator. As an application,
one gets isoperimetric inequalities on the 2-sphere for the third
eigenvalue of the Laplace Beltrami operator.

September

Monday September 14

Rafael Benguria (PUC)

The Brezis-Nirenberg Problem on S^n, in spaces of fractional dimension

Abstract: We consider the nonlinear eigenvalue problem,

-\Delta_{\mathbb{S^n}} u = \lambda u + |u|^{4/(n-2)} u,

with $u \in H_0^1(\Omega)$, where $\Omega$ is a  geodesic ball in S^n.
In dimension 3, this problem was considered by  Bandle and Benguria.
For positive radial solutions of this problem one is led to an
ordinary differential equation (ODE) that still makes sense when n is
a real rather than a natural number. Here we consider precisely that
situation with 2<n<4. Our main result is that in this case one has a
positive solution if and only if $\lambda \ge -n(n-2)/4$ is such that

\frac{1}{4} [(2 \ell_2 +1)^2 – (n-1)^2] < \lambda < \frac{1}{4} [(2
\ell_1 +1)^2 – (n-1)^2]

where $\ell_1$ (respectively $\ell_2$) is the first positive value of
$\ell$ for which the associated Legendre function ${\rm P}_{\ell}^{(2-n)/2} (\cos\theta_1)$ (respectively ${\rm P}_{\ell}^{(n-2)/2} (\cos\theta_1)$) vanishes.

Wednesday September 30

(Sala 1 de la Facultad de Matemáticas de la PUC)

16hrs.

Paolo Caldiroli (Universitá di Torino)

Isovolumetric and isoperimetric inequalities for a class of
capillarity functionals

Abstract: Capillarity functionals are parameter invariant functionals
defined on classes of two-dimensional parametric surfaces in
$\mathbb{R}^{3}$ as the sum of the area integral and an anisotropic
term of suitable form. In the class of parametric surfaces with the
topological type of the sphere and with fixed volume, extremals of
capillarity functionals are surfaces whose mean curvature is
prescribed up to a constant. For a certain class of anisotropies
vanishing at infinity, we prove existence and nonexistence of
volume-constrained, spherical-type, minimal surfaces for the
corresponding capillarity functionals. Moreover, in some cases, we
show existence of extremals for the full isoperimetric inequality.

17hrs.

Denis Bonhere (Université Libre de Bruxelles)

On the higher dimensional Extended Allen-Cahn equation

Abstract: In this talk, I will present results on a fourth order
extension of Allen-Cahn in a bounded domain of R^N with Navier
boundary conditions or in the whole space. The diffusion is driven by
a combination of the bilaplacian and the laplacian. In striking
contrast with the classical AC, establishing the sign and the symmetry
(when the domain is symmetric) of solutions minimizing the associated
functional is not an easy task. For bounded solutions in R^N, I will
present rigidity and Liouville type results and in particular an
analogue of the Gibbons’ conjecture.
The talk is based on a joint work with J. Földes & A. Saldaña and
another one with F. Hamel.

Radio Duna: Interview to Omar Gil

Omar Gil, screenwriter of “Primos entre sí”, the theater play sponsored by Capde, talks live in radio about the importance of bringing mathematics to the schoolrooms. He pointed out that this field of knowledge helps people to “understand better the world”. He also explains the need to accept mistakes during the process of learning.

August

Monday August 3

No seminar (Inverse Problems in the Physical Sciences, IP-Phys2015, see website

Monday August 10

16h

Didier Pilod (UFRJ, Brasil)

Dispersive perturbations of Burgers and hyperbolic equations

Monday August 24

16h

Claudio Muñoz (DIM-CMM)

Asymptotic Stability of solitons of the high dimensional Zakharov-Kuznetsov equation

In this talk I will discuss a recent work with R. Cote, D. Pilod and G. Simpson, where we considered solitons of the high dimensional Zakharov-Kuznetsov equation, a model of plasma ions in Physics. In particular, we prove that solitons are strongly asymptotically stable in the energy space, in a particular region of the plane determined by natural geometric and dispersive constraints. In proving this result we extend to the high dimensional case several tools coming from the one-dimensional setting (generalized KdV equations), introduced by Martel and Merle. However, some new difficulties arise when consider the dimension greater than two, in particular when proving required spectral properties, decay estimates, and the compactness in time of the asymptotic solution.

July

Monday July 6

16h

Jianfu Yang (TBA)

Equations involving fractional Laplacian operator: Compactness and applications

Abstract

17h

Huyuan Cheng (TBA)

Boundary blow-up solutions of fractional equations in a measure framework

Abstract

Monday July 13

16h

On the behaviour of solutions for the Novikov-Veselov equation

The Novikov-Veselov equation is a 2-dimensional generalization of the renowned Korteweg-de Vries equation integrable via the Inverse Scattering Transform (IST) for the 2-dimensional stationary Schrodinger equation. In this talk we will present recent results on the behaviour of solutions to the Novikov-Veselov equations: local well-posedness results, existence and absence of solitons, large-time behaviour, blow-ups etc. A part of results is obtained via IST techniques; for other results we make use of techniques of the theory of dispersive PDEs.

Monday July 27

No seminar (XVIII International Congress on Mathematical Physics, see website)

June

Monday June 15

16h

Dora Salazar (CMM)

Vortex-type solutions to a magnetic nonlinear Choquard equation

Abstract

17h

Sobre el problema de curvatura gaussiana prescrita

Abstract

May

Wednesday May 13, 12h, Sala Seminario 5to. piso. (Notar fecha y hora especiales) (Seminario anulado)

Yvan Martel (Ecole Polytechnique)

Survey on blow up for the critical generalized Korteweg-de Vries equation

We will review recent results with Frank Merle and Pierre Raphael (and also partly with Kenji Nakanishi) on blow up for critical generalized Korteweg-de Vries equation, and more generally on the classification of solutions close to the solitons.

Monday May 18, PUC, sala 1 Matematicas

16h

Jun Yang (Central China Normal University)

Vortex structures for maps from pseudo-Euclidean spaces

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three dimensional Euclidean space, we construct solutions with various vortex structures(vortex pairs, vortex circles and helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two dimensional elliptic problems with resolution theory given by the finite dimensional Lyapunov-Schmidt reduction method in nonlinear analysis.

16h50

Weiwei Ao (University of British Columbia)

Refined Finite-dimensional Reduction Method and Applications to Nonlinear Elliptic Equations

I will talk the refined finite dimensional reduction method and its application to nonlinear elliptic equations. We use this refined reduction method to get optimal bound on the number of interior spike solutions of the singularly perturbed Neumann problem as well as the boundary spike solutions. I will also talk about the entire solutions for nonlinear Schrodinger equations.

Monday May 25

15h:15

Ecuaciones de Sturm-Liouville singulares

Abstract

16h30

Eduardo Cerpa (UTFSM)

Control de la ecuación de Korteweg-de Vries

En esta charla introduciremos el concepto de controlabilidad de ecuaciones en derivadas parciales. En particular, estudiaremos el control frontera de la ecuación de Korteweg-de Vries y veremos cómo el dominio en donde la ecuación es estudiada puede influir en las propiedades del sistema. Nos interesaremos en algunos casos en donde la nolinealidad de la ecuación es crucial para demostrar su controlabilidad.

Wednesday May 27, 12h, Sala Seminario 5to. piso. (Notar fecha y hora especiales)

Marcelo Amaral (Unilab-Brazil)

Transmission problems on free interfaces

We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, L + and L -�, within their positive and negative sets respectively. A corresponding flux balance from one phase to another is also imposed. We establish existence and L infinity bounds of solutions. We also prove that variational solutions are non-degenerate and develop the regularity theory for solutions of such free boundary problems.

16th  March

15:00h

Mathematical study of a cell model for tumor growth : travelling front and incompressible limit

We consider mathematical models at macroscopic scale to describe tumor growth. In this view, tumor cells are considered as an elastic material subjected to mechanical pressure. Two main classes of model can be encountered: those describing the dynamics of tumor cells density and those describing the dynamic of the tumor thanks to the motion of its domain. These latter models are free boundary problem. We will show that such free boundary problem of Hele-Shaw type can be derived thanks to an incompressible limit from models describing the dynamics of cells density. Moreover, for this model we study the existence of travelling waves, allowing to describe the spread of the tumor.

16:05h

Pierpaolo Esposito (Universidad de Roma Tre)

Equilibria of point-vortices on closed surfaces

I will discuss the existence of equilibrium configurations for the Hamiltonian point-vortex model on a closed surface. Its topological properties determine the occurrence of three distinct situations, corresponding topologically to the sphere, to the real projective plane and to the remaining cases. As a by-product, new existence results are obtained for the singular mean-field equation with exponential nonlinearity.

Joint work with T. D’Aprile.

17:00h

Oscar Agudelo (University of West Bohemia)

Singularly perturbed Allen-Cahn equation with catenoidal nodal sets

In this lecture we review some recent results concerning existence and asymptotic behavior of solutions to the singularly perturbed problem

\alpha^2 \Delta u + u(1-u^2)=0, in Omega

where \Omega \subset \mathbb{R}^N is either a smooth bounded domain or the entire space and N\geq 2. We take advantage of the deep connection between the equation above and the theory of minimal surfaces to study asymptotic profiles of the solutions. Particular attention is paid to solutions with catenoidal nodal set.

From 30th March to 2nd April

Cursillo “On vortex dynamics in two-dimensional or three-dimensional incompressible flows”,

– Lecture 1 (Lunes 30 Marzo, 16h-17:30h, Sala Seminario 5to. piso):
Vortices in incompressible fluids.

In this first lecture we will consider the Euler equations governing the motion of incompressible fluids, in particular in a two-dimensional setting. We will focus on the vortex solutions and present Marchioro and Pulvirenti’s result on the derivation of the point vortex system from the 2D Euler equations. We will also briefly mention the vortex dynamics in other related equations (the Navier-Stokes equation and the Gross-Pitaevskii equation).

– Lecture 2 (Miércoles 1 Abril, 12h-13:30h, Sala Seminario 5to. piso)
Convergence of the point vortex system to the 2D Euler equation.

This lecture will explore more in detail the connection between the discrete model, described by the point vortex system, and the continuous fluid dynamics given by the Euler equation. We will show how the results by Goodman, Lowengrub and Hou and Schochet ensure that the point vortex system is a good approximation of the Euler equation when the number of vortices is large.

– Lecture 3 (Jueves 2 Abril, 14:30h-16h, Sala Seminario 7mo. piso):
Vortex filaments.

We will study the analogous notion of point vortices in three dimensions, namely the vortex filaments. We will explain the formal derivation leading to the binormal curvature flow equation governing the motion of one single vortex filament. We will also relate the binormal curvature flow equation and the cubic 1D Schrödinger equation via the Hasimoto transform. Finally we will present a system of simplified equations proposed by Klein, Majda and Damodaran to describe the interaction of several almost parallel vortex filaments.