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arXiv · 2205.09412

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

Davide Carazzato·Aldo Pratelli
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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.

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Davide Carazzato, Aldo Pratelli. 2022-05-19. On the existence and boundedness of minimizing measures for a general form of non-local energies. https://arxiv.org/abs/2205.09412

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