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 $\nabla\mathbf u \in BMO(\Omega;{ \mathbb R}^{ N\times n})$.
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