Calculus of Variations and Geometric Measure Theory
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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

[BibTeX]

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

ArXiv: 2002.07129 PDF
Links: Published Ahead of Print

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

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