Abstract: We discuss the regularity properties of two-dimensional stable s-minimal surfaces, presenting a robust regularity estimate and an optimal sheet separation bound, according to which the distance between different connected components of the surface must be at least the square root of 1-s.

Abstract: Suppose that two nonlocal minimal surfaces are included one into the other and touch at a point. Then, they must coincide. But this is perhaps less obvious than what it seems at first glance.

Abstract:  In this talk we discuss sharp versions of some spectral inequalities. The focus is on identifying optimal constants and extremal configurations, highlighting cases that are well understood, partially resolved, or still open to improvement.

Abstract:  In this talk I will show how optimal transport allows to identify Fokker-Planck equations for sticky-reflected Brownian motion (with boundary diffusion) as a Wasserstein gradient flow in the space of probability measures. The driving functional is a nonstandard relative entropy with respect to a reference measure, comprising an absolutely continuous interior part but also a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs with Wentzell boundary conditions, satisfying in addition an Energy Dissipation Inequality. If time permits we will also discuss a Schilder large-deviation microscopic origin of the problem, leading to a delicate phase transition when the boundary diffusion coefficient exceeds the interior one.

Joint work with Jean-Baptiste Casteras (CEMS Univ. Lisbon) and Filippo Santambrogio (Institut Camille Jourdain Univ. Lyon 1).

Abstract:  We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit PDE, in some specific cases. We will then spend some more time on a specific choice of extrapolation namely the metric extrapolation which appears to be a convex optimization problem.