DOI: 10.68381/jca19023 ISSN: 0944-6532
Characterization of the Multiscale Limit Associated with Bounded Sequences in BV
Rita Ferreira, Irene Fonseca
The notion of two-scale convergence for sequences of Radon measures with finite total variation is generalized to the case of multiple periodic length scales of oscillations. The main result concerns the characterization of
(n+1)
(
n
+
1
)
-scale limit pairs
(u,U)
(
u
,
U
)
of sequences
\{(u_\varepsilon{{\cal L}^N\!}_{\lfloor\Omega}, {Du_\varepsilon}_{\lfloor\Omega})\}_{\varepsilon>0}\subset {\cal M}(\Omega;\mathbb{R}^d)\times {\cal M}(\Omega; \mathbb{R}^{d\times N})
{
(
u
ε
L
N
⌊
Ω
,
D
u
ε
⌊
Ω
)
}
ε
>
0
⊂
M
(
Ω
;
R
d
)
×
M
(
Ω
;
R
d
×
N
)
whenever
\{u_\varepsilon\}_{\varepsilon>0}
{
u
ε
}
ε
>
0
is a bounded sequence in
BV(\Omega;\mathbb{R}^d)
B
V
(
Ω
;
R
d
)
. This characterization is useful in the study of the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space
BV
B
V
of functions of bounded variation and described by
n\in\mathbb{N}
n
∈
N
microscales, undertaken in another paper of the authors [“Reiterated homogenization in
BV
B
V
via multiscale convergence”, submitted].