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

2022/05/19 by Davide Carazzato, Carazzato, Davide, Aldo Pratelli +1 · 2 citations
Computer Science · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2205.09412

openalex publication_date 2022/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ 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^∞ functions, providing an a priori bound on their norm. We also derive a uniqueness result for minimizers.

Cited by

• Existence and nonexistence results for a nonlocal isoperimetric problem on ℍⁿ
• Continuity of critical points for 1-dimensional non-local energies

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