Adriano Prade et Andrea Natale

Adriano Prade (École Polytechnique)

Title: Young's law and regularity of minimizers for a nonlocal isoperimetric model of charged capillarity droplets.

Abstract: We consider a model for an electrically charged liquid drop laying on a solid surface. The surface tension of the droplet is represented by the classical De Giorgi perimeter with a capillarity modification whereas the electric charge is modeled by the Riesz energy. We study the resulting functional under convexity constraint in dimension 2, focusing on existence and regularity of minimizers. In the end we show the validity of Young's law, describing the contact angle between the droplet and the supporting surface.

Andrea Natale (Université Paris-Saclay/Inria PARMA)
Semi-discrete convex order and applications

Laguerre tessellations provide a computationally tractable way to describe a large number of convex partitions of Euclidean space. For this reason, they have become popular in computational geometry, imaging, and numerical analysis, both as a modeling and a discretization tool. In this talk, we consider an inverse problem arising from applications in imaging for material science: can we recover a Laguerre tessellation from the volumes and barycenters of its cells? We address this question by reformulating it as a Wasserstein projection onto the set of discrete measures in convex order with a given absolutely continuous measure. We provide a precise description of this set, establishing a direct connection with semidiscrete optimal transport theory. This enables us to develop a robust numerical method to compute Wasserstein projections under convex order constraints and construct approximate solutions to our tessellation reconstruction problem.
This talk is based on joint work with David Bourne, Thomas Gallouët and Quentin Mérigot.