In this note two results are established for energy functionals that are given by the integral of
W
(
x
,
∇
u
(
x
)
)
W({\mathbf x},\nabla {\mathbf u}({\mathbf x}))
over
Ω
⊂
R
n
\Omega \subset {\mathbb R}^n
with
∇
u
∈
B
M
O
(
Ω
;
R
N
×
n
)
\nabla {\mathbf u}\in \mathrm {BMO}(\Omega ;{\mathbb R}^{N\times n})
, the space of functions of Bounded Mean Oscillation of John and Nirenberg. A version of Taylor’s theorem is first shown to be valid provided the integrand
W
W
has polynomial growth. This result is then used to demonstrate that every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the second variation of the energy is uniformly positive is a strict local minimizer of the energy in
W
1
,
B
M
O
(
Ω
;
R
N
)
W^{1,\mathrm {BMO}}(\Omega ;{\mathbb R}^N)
, the subspace of the Sobolev space
W
1
,
1
(
Ω
;
R
N
)
W^{1,1}(\Omega ;{\mathbb R}^N)
for which the weak derivative
∇
u
∈
B
M
O
(
Ω
;
R
N
×
n
)
\nabla {\mathbf u}\in \mathrm {BMO}(\Omega ;{\mathbb R}^{N\times n})
.