08:30-09:30 Wednesday, 2 April 2025 Auditorium, Rectorate
Abstract
Nonlocal shape optimization problems involving interaction energies with competing repulsive and attractive terms arise in a variety of applications and have been extensively studied in the mathematical community over the past decades. In this talk I will consider a family of nonlocal energies defined on sets with prescribed mass, where the repulsive interaction is an anisotropic variant of the Coulomb kernel and the attractive interaction is quadratic. Under the sole assumption that the Fourier transform of the anisotropic kernel is strictly positive, we establish the existence of a critical mass threshold: above this value, ellipsoids are the unique minimizers, while below it, minimizers fail to exist.If instead the Fourier transform is merely nonnegative, we observe a dichotomy: either a critical mass exists as in the previous case, or ellipsoids are minimizers for all mass values. The presence or absence of a critical mass is related to the shape of minimizers when the energy is considered on the broader class of measures with prescribed mass.This is a joint work with Riccardo Cristoferi (Radboud University) and Lucia Scardia (Heriot-Watt University).