Calculus of Variations
& Geometric Analysis

Title: First examples of sharp quantitative Talenti’s inequality 

Abstract:   A classical result by Talenti provides a comparison for solutions of Elliptic PDE with Dirichlet boundary conditions. In particular, it implies that the L^p-norm of the solution to -Δu = f in a domain Ω is smaller than the L^p-norm of the solution to -Δv=f^# where f^# is the symmetric decreasing rearrangement of f.

Recently, [Amato, Barbato, Lia Masiello, Paoli 2023] proved a quantitative version of this result, but only in the case p=+∞, and with a non-sharp exponent.

In this talk, we aim at obtaining a similar result, but for any p and with a sharp exponent. To that end we make the extra-assumption that Ω is a ball B (therefore Ω is already symmetric and only f is symmetrized). We start by analyzing the case where f is the characteristic function of a set E ⊆ B and show that in this case, one can reach a sharp quantitative estimate, using a shape derivative approach. We will then explain how one can tackle the more general case of a general bounded function f.

Title: Shape optimization under thickness constraint

Abstract: Among the various width-type constraints considered in shape optimization, the most classical is the maximal width (or diameter), which describes how "large" an admissible set can be. Less commonly studied - but equally significant - is the minimal width, which instead measures the "thickness" of admissible shapes.

In this seminar, I will present some recent results on the optimization, in quantitative form, of the area and the Cheeger constant for sets with prescribed thickness. The talk is based on joint works with Davide Zucco (Turin, Italy), Giorgio Saracco (Ferrara, Italy), and Ilias Ftouhi (Nîmes, France)..

Title: Quantitative Stability of the push-forward operation by an optimal transport map 

Abstract: We study the quantitative stability of the mapping that to a measure associates its push-forward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.

From a joint work with:  Guillaume Carlier and Alex Delalande

Title: TBA

Abstract: TBA