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.