  In this note two results are established for energy functionals that are
given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over
$\Omega \subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(\Omega;{\mathbb
R}^{N\times n})$, the space of functions of Bounded Mean Oscillation of John &
Nirenberg. A version of Taylor's theorem is first shown to be valid provided
the integrand $W$ has polynomial growth. This result is then used to
demonstrate that, for the Dirichlet, Neumann, and mixed problems, 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 $W^{1,BMO}(\Omega;\mathbb{R}^N)$, the subspace
of the Sobolev space $W^{1,1}(\Omega;\mathbb{R}^N)$ for which the weak
derivative $\nabla\mathbf u \in BMO(\Omega;{\mathbb R}^{N\times n})$.
