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

Qinglan Xia and Bohan Zhou ORCID logo

Abstract

In this article, we consider the (double) minimization problem

min{P(E;Ω)+λWp(E,F):EΩ,FRd,|EF|=0,|E|=|F|=1},

where λ0, p1, Ω is a (possibly unbounded) domain in Rd, P(E;Ω) denotes the relative perimeter of 𝐸 in Ω and Wp denotes the 𝑝-Wasserstein distance. When Ω is unbounded and d3, 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 the dimension d1, 1p+2d>1, Ω=Rd and 𝜆 is sufficiently small.

MSC 2010: 49J45; 49Q20; 49Q05; 49J20

  1. Communicated by: Frank Duzaar

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Received: 2020-08-09
Revised: 2020-12-10
Accepted: 2021-02-18
Published Online: 2021-03-13

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