Balls Are Maximizers of the Riesz-type Functionals with Supermodular Integrands

Over the last decades, one field of intense research activity has been the study of extremals of integral functionals. The Riesz-type kind has attracted growing attention and played a crucial role in the resolution of Choquard’s conjecture in a breakthrough paper by E. H. Lieb [1]. The determination of cases of equality in the Riesz-rearrangement inequality has also received a large amount of interest from mathematicians due to its connection with many other functional inequalities and its several applications to physics [2, 3, 4]. Variational problems for steady axisymmetric vortex-rings in which kinetic energy is maximized subject to prescribed impulse involves Riesztype functionals with constraints. In [5], G. R. Burton has proved the existence of maximizers in an extended constraint set, he has also showed that the maximizer is Schwarz symmetric (up to translations). His method hinges on a resolution of an optimization of a Riesz-type functional under constraint [5, Proposition 8]. The purpose of this paper is to answer the more general question: When do maximizers of the Riesz-type functional inherit the symmetry and monotonicity properties of the integrand involved in it? The method of G. R. Burton [5] cannot apply to solve the above problem. In this paper, we develop a self-contained approach. Let us give here a foretaste of our ideas. First, we recall that: A Riesz-type functional is a functional of the form: