The lecture is devoted to a review of some stability results in simple interpolation inequalities, typically Gagliardo-Nirenberg-Sobolev inequalities. When optimal constants are known and optimal functions are characterized, it is a natural question to ask whether the deficit, that is, the difference of the two sides of the inequality, controls a distance to the set of optimal functions. After...
Optimization problems on probability measures in $\mathbb{R}^d$ are considered where the cost functional involves multi-marginal optimal transport. In a model of $N$ interacting particles, the interaction cost is repulsive and described by a two-point function $c(x,y) =\ell(|x-y|)$ where $\ell: \mathbb{R}_+ \to [0,\infty]$ is decreasing to zero at infinity. Due to a possible loss of mass at...
In this talk, I will first state a general result about the uniqueness and the non-degeneracy of positive radial solutions to some semi-linear elliptic equations $-\Delta u=g(u)$. Then I will consider the case of the double power non-linearity $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$. In this case, the non-degeneracy of the unique solution $u_\mu$ allows us to derive its behavior in...
The mean curvature flow plays an important role in many applications in physics or numerical engineering, for example in image processing or material science. The talk will focus on new neural network-based numerical methods for approximating the mean curvature flow of general interfaces, both oriented and non-oriented. To learn the correct evolution law, our networks are trained on implicit...
(joint work with Jean Cauvin-Vila and Amaury Hayat)
This work is motivated by a collaboration with the French Photovoltaic Institute. The aim of the project is to propose a model in order to simulate and optimally control the fabrication process of thin film solar cells. The production of the thin film inside of which occur the photovoltaic phenomena accounting for the efficiency of the...
