We study the relaxation with respect to the
L^1
L
1
norm of integral functionals of the type
F(u)=\int_\Omega f(x,u,\nabla u)\,dx\,\quad u\in W^{1,1}(\Omega;S^{d-1})
F
(
u
)
=
∫
Ω
f
(
x
,
u
,
∇
u
)
d
x
u
∈
W
1
,
1
(
Ω
;
S
d
−
1
)
where
\Omega
Ω
is a bounded open set of
R^N
R
N
,
S^{d-1}
S
d
−
1
denotes the unite sphere in
R^d
R
d
,
N
N
and
d
d
being any positive integers, and
f
f
satisfies linear growth conditions in the gradient variable. In analogy with the unconstrained case, we show that, if, in addition,
f
f
is quasiconvex in the gradient variable and satisfies some technical continuity hypotheses, then the relaxed functional
\overline F
F
‾
has an integral representation on
BV(\Omega;S^{d-1})
B
V
(
Ω
;
S
d
−
1
)
of the type
\bar F(u)=\int_{\Omega}f(x,u,\nabla u)\,dx+\int_{S(u)}K(x,u^-,u^+,\nu_u)\,d{\cal H}^{N-1} + \int_\Omega f^\infty (x,u,d C(u)),
F
ˉ
(
u
)
=
∫
Ω
f
(
x
,
u
,
∇
u
)
d
x
+
∫
S
(
u
)
K
(
x
,
u
−
,
u
+
,
ν
u
)
d
H
N
−
1
+
∫
Ω
f
∞
(
x
,
u
,
d
C
(
u
)
)
,
where the suface energy density
K
K
is defined by a suitable Dirichlet-type problem.