Overview
This study investigates a one-parameter family of nonlocal energies defined on the real line. These energy functionals are applied to scalar functions characterized by derivatives that manifest as a periodic arrangement of screened Dirac masses. The primary finding establishes that equi-spaced particle configurations serve as the global minimizers for these specific energy functionals.
Research Context
The research focuses on a class of nonlocal energies that encompasses subcritical Gagliardo seminorms and Riesz functionals. These functionals are applied within a one-dimensional setting to scalar functions. A key characteristic of these scalar functions is that their derivatives are composed of a periodic configuration of screened Dirac masses. The overarching goal is to identify the global minimizers within this defined system.
Approach
The methodology employed to achieve the result involved representing the energy functionals as interaction potentials. These interaction potentials are specifically dependent on the geodesic distances separating the particles. The approach further leveraged the complete monotonicity properties inherent in these interaction potentials. A significant component of the proof relies on invoking the Cohn-Kumar Universality Theorem.
Findings
The study determined that for the introduced one-parameter family of nonlocal energies, the global minimizers are precisely the equi-spaced configurations of particles. This finding is applicable to energy functionals that include subcritical Gagliardo seminorms and Riesz functionals, when acting on scalar functions whose derivatives consist of periodic screened Dirac masses.