Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers

Abstract

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

8 pages