On the existence and boundedness of minimizing measures for a general form of non-local energies

TL;DR

This paper establishes the existence, boundedness, and uniqueness of minimizing measures for a broad class of non-local energies defined on Radon measures, extending traditional results to more general settings.

Contribution

It introduces a general framework for non-local energies on Radon measures and proves existence, boundedness, and uniqueness of minimizers, including cases where minimizers are bounded functions.

Findings

Existence of optimal measures in a broad non-local energy framework

Optimal measures are often bounded functions with an a priori norm bound

Uniqueness of minimizers under certain conditions

Abstract

In this paper we consider a very general form of a non-local energy in integral form, which covers most of the usual ones (for instance, the sum of a positive and a negative power). Instead of admitting only sets, or $L^\infty$ functions, as admissible objects, we define the energy for all the Radon measures. We prove the existence of optimal measures in a wide generality, and we show that in several cases the optimal measures are actually $L^\infty$ functions, providing an a priori bound on their norm. We also derive a uniqueness result for minimizers.