论文标题
在无限域中，Wasserstein惩罚项的同类问题的存在最小化器的存在
The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
论文作者
论文摘要
在本文中，我们考虑（双重）最小化问题$$ \ min \ weft \ { f \ rvert = 1 \ right \}，$$其中$ p \ geqslant 1 $，$ω$是一个（可能是无限的）域中的$ \ mathbb {r}^d $，$ p（e;ω）$表示相对的$ e $ in $ e $ in $ f.p $ w_p $ decess $ p $ p-p pecast $ e $ e $ - 当$ω$不受限制和$ d \ geqslant 3 $时，这是Buttazzo，Carlier和Laborde在纸质奇异措施之间的Wasserstein距离上提出的一个开放问题。当$ \ frac {1} {p}+\ frac {2} {2} {d}> 1 $，$ω= \ mathbb {r}^d $和$λ$时，我们证明了这个问题的最小化存在。
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.
