论文标题
泰勒（Taylor）在BMO上的功能定理，并应用于BMO本地最小化器
Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers
论文作者
论文摘要
在本说明中，为能量功能建立了两个结果，这些功能由$ w（\ Mathbf x，\ nabla \ Mathbf U（\ Mathbf x））$ ofum $ ws $ω\ subset \ subset \ subset \ mathbb {r}^n $与$ \ nabla \ nabla \ nabla \ mathbf U \ in bmath in time n ocy（\ mathbf x））上约翰＆Nirenberg的有限平均振荡功能空间。泰勒定理的一个版本首先证明是有效的，只要intaptand $ w $具有多项式增长。然后使用该结果证明，对于dirichlet，neumann和混合问题，每个lipschitz-连续的解决方案是相应的欧拉 - 拉格朗日方程式，在该方程中，能量的第二个变化是均匀积极的，这是$ w^^{1，bmo}（1，bmo}（ω; \ mathbbbb {subbb {subbb {subbb {subbbe $ w^{1,1}（ω; \ mathbb {r}^n）$，bmo（ω; {\ mathbb r}^{n \ times n} in bmo（ω; {\ mathbb r}^{n}）$弱衍生物$ \ nabla \ mathbf u \ $。
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 $\nabla\mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$.
