BMO上泛函的Taylor定理及其对BMO局部极小的应用
Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers
摘要：
本文建立了由$ W(\mathbf x,\nabla \mathbf u(\mathbf x))$在$Ω\subset\mathbb{R}^n$上与$\nabla \mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$的积分给出的能量泛函的两个结果，即John & Nirenberg的有界平均振荡函数空间.泰勒定理的一个版本首先被证明是有效的被积函数$W$有多项式增长。这个结果然后被用来证明，对于Dirichlet，Neumann，和混合问题，每一个相应的欧拉-拉格朗日方程的Lipschitz连续的解决方案，在第二阶变分的能量是一致积极的是一个严格的局部极小的能量在$W^{1,BMO}(Ω;\mathbb{R}^N)$，索伯列夫空间$W^{1,1}(Ω;\mathbb{R}^N)$的子空间，其中弱导数$\nabla\mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$。
英文摘要：
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 $Ω\subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$, 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 $W^{1,BMO}(Ω;\mathbb{R}^N)$, the subspace of the Sobolev space $W^{1,1}(Ω;\mathbb{R}^N)$ for which the weak derivative $\nabla\mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$.
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