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research
Multiscale homogenization of convex functionals with discontinuous integrand
Authors
Marco Barchiesi
Publication date
1 January 2006
Publisher
View
on
arXiv
Abstract
This article is devoted to obtain the
Γ
\Gamma
Γ
-limit, as
ϵ
\epsilon
ϵ
tends to zero, of the family of functionals
F
ϵ
(
u
)
=
∫
Ω
f
(
x
,
x
ϵ
,
.
.
.
,
x
ϵ
n
,
∇
u
(
x
)
)
d
x
F_{\epsilon}(u)=\int_{\Omega}f\Bigl(x,\frac{x}{\epsilon},..., \frac{x}{\epsilon^n},\nabla u(x)\Bigr)dx
F
ϵ
​
(
u
)
=
∫
Ω
​
f
(
x
,
ϵ
x
​
,
...
,
ϵ
n
x
​
,
∇
u
(
x
)
)
d
x
, where
f
=
f
(
x
,
y
1
,
.
.
.
,
y
n
,
z
)
f=f(x,y^1,...,y^n,z)
f
=
f
(
x
,
y
1
,
...
,
y
n
,
z
)
is periodic in
y
1
,
.
.
.
,
y
n
y^1,...,y^n
y
1
,
...
,
y
n
, convex in
z
z
z
and satisfies a very weak regularity assumption with respect to
x
,
y
1
,
.
.
.
,
y
n
x,y^1,...,y^n
x
,
y
1
,
...
,
y
n
. We approach the problem using the multiscale Young measures.Comment: 18 pages; a slight change in the title; to be published in J. Convex Anal. 14 (2007), No.
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Archivio istituzionale della ricerca - Università di Trieste
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Last time updated on 16/10/2019