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A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case
Authors
Giuliano Gargiulo
Elvira Zappale
Publication date
24 May 2023
Publisher
View
on
arXiv
Abstract
In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type
W
1
,
β
(
Ξ©
;
R
d
)
β
u
β¦
sup
β‘
e
s
s
(
x
,
y
)
β
Ξ©
W
(
x
,
y
,
β
u
(
x
)
,
β
u
(
y
)
)
,
W^{1,\infty}(\Omega;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in \Omega} W(x,y, \nabla u(x),\nabla u(y)),
W
1
,
β
(
Ξ©
;
R
d
)
β
u
β¦
sup
ess
(
x
,
y
)
β
Ξ©
β
W
(
x
,
y
,
β
u
(
x
)
,
β
u
(
y
))
,
where
Ξ©
\Omega
Ξ©
is a bounded open subset of
R
N
\mathbb R^N
R
N
and
W
:
Ξ©
Γ
Ξ©
Γ
R
d
Γ
N
Γ
R
d
Γ
N
β
R
W:\Omega \times \Omega \times \mathbb R^{d \times N}\times \mathbb R^{d \times N} \to \mathbb R
W
:
Ξ©
Γ
Ξ©
Γ
R
d
Γ
N
Γ
R
d
Γ
N
β
R
.Comment: to appear in European Journal of Mathematic
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oai:arXiv.org:2305.14801
Last time updated on 26/05/2023