*Published Paper*

**Inserted:** 15 mar 2021

**Last Updated:** 1 sep 2021

**Journal:** Advances in Calculus of Variations

**Year:** 2021

**Doi:** https://doi.org/10.1515/acv-2020-0083

**Abstract:**

In this article, we consider the (double) minimization problem

$\min\left\{P(E;\Omega)+\lambda W_p(E,F):~E\subseteq\Omega,~F\subseteq \mathbb{R}^d,~\lvert E\cap F\rvert=0,~ \lvert E\rvert=\lvert F\rvert=1\right\},$

where $\lambda\geqslant 0$, $p\geqslant 1$, $\Omega$ is a (possibly unbounded) domain in $\mathbb{R}^d$, $P(E;\Omega)$ denotes the relative perimeter of $E$ in $\Omega$ and $W_p$ denotes the $p$-Wasserstein distance. When $\Omega$ is unbounded and $d\geqslant 3$, it is an open problem proposed by Buttazzo, Carlier and Laborde in the paper \textit{On the Wasserstein distance between mutually singular measures}. We prove the existence of minimizers to this problem when the dimension $d\geqslant 1$, $\frac{1}{p}+\frac{2}{d}>1$, $\Omega=\mathbb{R}^d$ and $\lambda$ is sufficiently small.

**Keywords:**
Wasserstein distance, unbounded domains, isoperimetric problem, Volume constraints, quasi-perimeter