Titres et résumés

Bohdan Bulanyi (Paris 7): "Regularity of a minimizers in 2d of the optimal p-compliance problem with length penalization"

The presentation will focus on "some results" recently obtained as a part of my thesis, about the regularity of a minimizers in 2d of the optimal p-compliance problem with length penalization.

Philippe Castillon (Montpellier): "Prescribing the Gauss curvature of hyperbolic convex bodies"

The Gauss curvature measure of Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Given a measure $\mu$, Alexandrov’s problem consists in finding a convex body whose curvature measure is $\mu$. In Euclidean space, A.D. Alexandrov gave necessary and sufficient conditions on $\mu$ for this problem to have a solution, and it was observed later that proving the existence of a convex body of given curvature measure $\mu$ is equivalent to an optimal transport problem on the sphere.

In this talk I will address Alexandrov’s problem for arbitrary convex bodies in the hyperbolic space. After defining the curvature measure, I will give the necessary and sufficient conditions on a measure $\mu$ to be the curvature of a convex body, and I will explain how the optimal transport approach leads to a non-linear Kantorovich problem on the sphere which is the weak form of a Monge-Ampère equation.

Alexandre Delyon (Paris 6): "Understanding the eggs of branchiopods"

Eulimnadiae are little shrimps living in ephemeral water puddles. Their specificity lies in the shape of their eggs, that is very uncommon to the eyes of biologist. Evolution theory allows us to believe that their shape is the result of a long process of optimization, whose criteria are unknown for the moment. Our goal is to propose criteria for which the shape of the eggs is optimal. Indeed, this would give an explanation to the biologist. In the talk we propose the following criterium: A shrimp wants to maximize the number of eggs in the uterus with some constraints. This leads to a packing problem that we investigate.

Guido De Philippis (Trieste): "Fine structure of measures satisfying a PDE constraint"

In this talk I will present some new result concerning the structure of measure satisfying a linear PDE constraint. In 2016, in collaboration with Filip Rindler, we prove a first structural result concerning the singular part of measure subject to PDE constraint. This turned out to have several applications in GMT and in Geometric Analysis. Recently, in a joint work with Adolfo Arroyo Rabasa, Jonas Hirsch and Filip Rindler we improve upon this result proving a more precise structure on the “low” dimensional part of the measure. As a corollary we recover several known rectifiability results. In this talk I will try to give an overview of both these results and of their applications.

Jimmy Lamboley (Paris 6): "Blaschke-Santalo diagram and eigenvalues"

Given three shape functionals (associating a real number to each set in $\mathbb{R}^n$), one can seek to describe all possible inequalities involving these three functionals, for a certain class of domains. This study rely on the description of the so-called Blaschke-Santalo diagram (see below for an example). These questions have been extensively studied for purely geometrical functionals, and for planar convex domains, even though some questions remain open. The goal here is to start extending these studies to spectral type functionals. Denoting $\lambda_1$ the first Dirichlet eigenvalue, $P$ the perimeter, and $|\cdot|$ the volume, we are for example interested in describing the following set:

Dario Mazzoleni (Brescia): "Asymptotic spherical shapes in spectral optimization problems"

We study the positive principal eigenvalue of a weighted problem associated with the Neumann-Laplacian settled in a box $\Omega\subset \R^N$, which arises from the investigation of the survival threshold in population dynamics.
When trying to minimize such eigenvalue with respect to the sign-changing weight, one is lead to consider a shape optimization problem, which is known to admit spherical optimal shapes only in trivial cases.
We investigate if spherical shapes can be recovered in the limit when the negative part of the weight diverges.
First of all, we show that the shape optimization problem appearing in the limit is the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian.
Thanks to $\alpha$-symmetrization techniques on cones, it can be proved that optimal shapes for the spectral drop problem are spherical for suitable choices of the box, the most interesting case being when $\Omega$ is a convex polytope, and in this case a quantitative analysis of the convergence can be performed.
Finally, for a smooth $\Omega$, we show that small volume spectral drops are asymptotically spherical, with center at points with high mean curvature. This is a joint project with Benedetta Pellacci and Gianmaria Verzini.