The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
英文摘要：
In this article, we consider the (double) minimization problem $$\min\left\{P(E;Ω)+λW_p(E,F):~E\subseteqΩ,~F\subseteq \mathbb{R}^d,~\lvert E\cap F\rvert=0,~ \lvert E\rvert=\lvert F\rvert=1\right\},$$ where $p\geqslant 1$, $Ω$ is a (possibly unbounded) domain in $\mathbb{R}^d$, $P(E;Ω)$ denotes the relative perimeter of $E$ in $Ω$ and $W_p$ denotes the $p$-Wasserstein distance. When $Ω$ 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 $\frac{1}{p}+\frac{2}{d}>1$, $Ω=\mathbb{R}^d$ and $λ$ is sufficiently small.
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