All posts by jdavila






Monday 16


16 hrs.

Matteo Rizzi (SISSA, Italy)

Clifford Tori and the singularly perturbed Cahn-Hilliard equation




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




Monday 30


Nicolás Carreño (USM)

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




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.


Wednesday October 14


Søren Fournais (Aarhus University)


Optimal magnetic Sobolev constants in the semiclassical limit

Abstract Soeren



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



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 $R^3$.


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.



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)


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.


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.