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.