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

TL;DR

This paper extends Taylor's theorem to energy functionals with integrands of polynomial growth on BMO spaces, and shows that solutions with positive second variation are strict local minimizers in the BMO Sobolev space.

Contribution

It establishes a version of Taylor's theorem for functionals on BMO and applies it to prove local minimality of solutions with positive second variation.

Findings

Taylor's theorem valid for BMO-based functionals with polynomial growth

Lipschitz solutions with positive second variation are strict local minimizers in BMO Sobolev space

Results apply to Dirichlet, Neumann, and mixed boundary problems

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