The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains (Q2677297)

In this paper, the authors consider a problem left open by \textit{G. Buttazzo} et al. [Adv. Calc. Var. 13, No. 2, 141--154 (2020; Zbl 1446.49010)]. They prove an existence result for a double minimization problem of type perimeter-Wesserstein on unbounded domains. More precisely, let \(p\ge 1\) and \(d\ge 1\) with \[ \frac{1}{p}+\frac{2}{d}>1. \] Then, the following existence result holds true: there exists \(\lambda_0=\lambda_0(d,p)\) such that for each \(\lambda\in (0,\lambda_0]\) the minimization problem \[ \min\{P(E)+\lambda W_p(\mathcal L^d _E,\mathcal L^d_F) : E,F\subseteq \mathbb R^d,\,\mathcal L^d(E\cap F)=0,\,\mathcal L^d(E)=\mathcal L^d(F)=1\} \] has a solution. Here, \(P\) is the perimeter, \(W_p\) is the \(p\)-Wesserstein distance and \(\mathcal L^d_X\) denotes the \(d\)-dimensional Lebesgue measure restricted to \(D\).