The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains

Abstract

In this article, we consider the (double) minimization problem $$\min \left\{P(E;\hat{I}\text{c◯})+\hat{I}»W_p(E,F):E\subseteq \hat{I}\text{c◯},F\subseteq {\mathbb{R}}^d,|E\cap F|=0,|E|=|F|=1\right\},$$ where $p⩾1$, $\hat{I}\text{c◯}$ is a (possibly unbounded) domain in ${\mathbb{R}}^d$, $P(E;\hat{I}\text{c◯})$ denotes the relative perimeter of $E$ in $\hat{I}\text{c◯}$ and $W_p$ denotes the $p$-Wasserstein distance. When $\hat{I}\text{c◯}$ is unbounded and $d⩾3$, it is an open problem proposed by Buttazzo, Carlier and Laborde in the paper ON THE WASSERSTEIN DISTANCE BETWEEN MUTUALLY SINGULAR MEASURES. We prove the existence of minimizers to this problem when $\frac{1}{p}+\frac{2}{d}>1$, $\hat{I}\text{c◯}={\mathbb{R}}^d$ and $\hat{I}»$ is sufficiently small.