I will construct an example of a bounded planar domain with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This shows that Payne's nodal line conjecture can at most hold for simply-connected domains in the plane.

The classical Stein-Tomas theorem extends from the theory of linear Fourier restriction estimates for smooth manifolds to the one of fractal measures exhibiting Fourier decay. In the multilinear “smooth” setting, transversality allows for estimates beyond those implied by the linear theory. The goal of this talk is to investigate the question “how does transversality manifest itself in the fractal world?” We will show, for instance, that it could be through integrability properties of the multiple convolution of the measures involved, but that is just the beginning of the story. In the special case of Cantor-type fractals, we will construct multilinear Knapp examples through certain co-Sidon sets which, in some cases, will give more restrictive necessary conditions for a multilinear theorem to hold than those currently available in the literature. This is work in progress with Ana de Orellana (University of St. Andrews, Scotland).

We will address the problem of assigning optimal routes in a graph that transport two given densities over the nodes. The occupation of each edge at a given time defines a metric over this graph, for which the routes must be geodesics. This model may describe for example the congestion of a city and its solutions are known as Wardrop equilibria. Additionally, a central planner can require that the assignment is efficient, meaning it minimizes the Kantorovich functional arising from this metric. In this presentation, we will characterize this problem in terms of a partial differential equation and illustrate a simple case. This work is a collaboration with Sergio Zapeta Tzul, a former MSc student at CIMAT and current PhD student at the University of Minnesota.

I will present some integral representation results for the lower semicontinuous envelope of integral functionals defined in the space of functions with generalized bounded variation.

The talk is based on joint projects with Lorenza D'Elia, Giacomo Bertazzoni, Petteri Harjulehto and Peter Hasto.

We consider the Cauchy problem related to the family of $k$-dispersion generalized Benjamin-Ono ($k$-DGBO) equations:
\begin{equation}\label{DGBOINTRO}
\begin{cases}
u_t + D_x^\alpha u_x + \mu u^ku_x= 0, \quad (t,x) \in \mathbb{R} \times \mathbb{R},\\
u(0,x)=u_0(x),
\end{cases}
\end{equation} where $u = u(t,x)$ is real-valued, $\alpha \in [1,2]$, $\mu \in \{\pm 1\}$ and $k \in \mathbb{Z}^+$. Here, ${D^\alpha_x}$ represents the 1-dimensional fractional Laplacian operator in the spatial variable $x$. For $k \geq 4$, we establish local and global well-posedness results for \eqref{DGBOINTRO} in both the critical $\left(s= \frac{k-2 \alpha}{2k}\right)$ and subcritical $\left(s > \frac{k-2 \alpha}{2k}\right)$ regimes, addressing sharp regularity in homogeneous and inhomogeneous Sobolev spaces. Additionally, our method enables the formulation of a scattering criterion and a scattering theory for small data. We also investigate the case $k = 3$ via frequency-restricted estimates, obtaining local well-posedness results for the initial value problem associated with the $3$-DGBO equation and generalizing the existing results in the literature for the whole subcritical range. For higher dispersion, these local results can be extended globally even for rough data, particularly for initial data in Sobolev spaces with negative indices. As a byproduct, we derive new nonlinear smoothing estimates. This is a joint work with Luccas Campos (UFMG) and Felipe Linares (IMPA).