Existence and stability results for an isoperimetric problem with a non-local interaction of Wasserstein type

TL;DR

This paper proves the existence of minimizers for a variational problem combining perimeter and Wasserstein-type non-local energy, and shows that balls uniquely minimize the energy when perimeter dominates.

Contribution

It extends previous results to the full parameter range and establishes existence and uniqueness of minimizers in certain regimes.

Findings

Existence of minimizers for the combined perimeter and Wasserstein energy.

Uniqueness of ball minimizers when perimeter dominates.

Extension of previous partial results to the full parameter range.

Abstract

The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial results to the full range of parameters. We also show that in the regime where the perimeter is dominant, the energy is uniquely minimized by balls.