Taylor’s theorem for functionals on BMO with application to BMO local minimizers

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}) .