Calculus of Variations and Geometric Measure Theory

Q. Xia - B. Zhou

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

created by zhou on 15 Mar 2021
modified by xia1 on 01 Sep 2021


Published Paper

Inserted: 15 mar 2021
Last Updated: 1 sep 2021

Journal: Advances in Calculus of Variations
Year: 2021

ArXiv: 2002.07129 PDF
Links: Published Ahead of Print


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