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.