We give a first variation formula for functionals of the type
∫
Ω
f
(
x
,
μ
)
\int _\Omega {f(x,\mu )}
, where
f
(
x
,
p
)
:
Ω
×
R
k
→
R
f(x,p):\Omega \times {{\mathbf {R}}^k} \to {\mathbf {R}}
is of linear growth in
p
p
for large
|
p
|
|p|
and
μ
\mu
is a
R
k
{{\mathbf {R}}^k}
-valued measure in
Ω
\Omega
. The Euler equation for the minima of various functionals defined on spaces of
BV
{\text {BV}}
functions is then studied.