A lower semicontinuity result for some integral functionals in the space SBD
Abstract
The purpose of this paper is to study the lower semicontinuity with respect to the strong
L
1
-convergence, of some integral functionals defined in the space SBD of special functions with bounded deformation. Precisely, let
U
be a bounded open subset of
R
n
. If
u∈
SBD
(U)
,
(
u
h
)⊂
SBD
(U)
converges to
u
strongly in
L
1
(U,
R
n
)
and the measures
|
E
j
u
h
|
converge weakly * to a measure
ν
singular with respect to the Lebesgue measure, then
∫
U
f(x,Eu)dx≤
lim inf
h→∞
∫
U
f(x,E
u
h
)dx
provided
f
satisfies some weak convexity property and the standard growth assumptions of order
p>1
.