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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 EMAIL logo


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

min { P ( E ; Ω ) + λ W p ( E , F ) : E Ω , F R d , | E F | = 0 , | E | = | F | = 1 } ,

where λ 0 , p 1 , Ω is a (possibly unbounded) domain in 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 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 the dimension d 1 , 1 p + 2 d > 1 , Ω = R d 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
Published in Print: 2023-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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