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 $$\min\left{P(E;\Omega)+\lambda W_p(E,F):~E\subseteq\Omega,~F\subseteq \mathbb{R}^d,~\lvert E\cap F\rvert=0,~ \lvert E\rvert=\lvert F\rvert=1\right},$$ where $p\geqslant 1$, $\Omega$ 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 ON THE WASSERSTEIN DISTANCE BETWEEN MUTUALLY SINGULAR MEASURES. We prove the existence of minimizers to this problem when $\frac{1}{p}+\frac{2}{d}>1$, $\Omega=\mathbb{R}^d$ and $\lambda$ is sufficiently small.