# All posts by jdavila

# November

**Monday 16**

*16 hrs.*

*Matteo Rizzi (SISSA, Italy)*

*Clifford Tori and the singularly perturbed Cahn-Hilliard equation*

*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*

*16hrs.*

**Felipe Barra (DFI, U. de Chile)***Termodinámica de sistemas cuánticos abiertos.*

# October

**Wednesday October 14**

*16hrs.*

**Søren Fournais (Aarhus University)**

*Optimal magnetic Sobolev constants in the semiclassical limit*

*17hrs.*

*Matteo Cozzi (University of Milan)*

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

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.