Affiliation:
1. Department of Mathematics , University of California at Davis , Davis , California , USA
2. Department of Mathematics , Dartmouth College , Hanover , New Hampshire , USA
Abstract
Abstract
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
}
,
\min\{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\},
where
λ
⩾
0
\lambda\geqslant 0
,
p
⩾
1
p\geqslant 1
, Ω is a (possibly unbounded) domain in
R
d
\mathbb{R}^{d}
,
P
(
E
;
Ω
)
P(E;\Omega)
denotes the relative perimeter of 𝐸 in Ω and
W
p
W_{p}
denotes the 𝑝-Wasserstein distance.
When Ω is unbounded and
d
⩾
3
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 the dimension
d
⩾
1
d\geqslant 1
,
1
p
+
2
d
>
1
\frac{1}{p}+\frac{2}{d}>1
,
Ω
=
R
d
\Omega=\mathbb{R}^{d}
and 𝜆 is sufficiently small.
Subject
Applied Mathematics,Analysis
Cited by
4 articles.
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