关于尖链在迁移定理中的作用
On the role of sharp chains in the transport theorem
摘要：
在Federer几何测度理论的基础上，给出了对流不规则区域的一个广义迁移定理。一个典型的$r$维域被看作是一个$n$维欧氏空间的开集上的有限质量的平坦$r$链。这样一个广义域在时间上的演化被假定为是按照一个bi-Lipschitz型映射。所示的诱导曲线是连续的，相对于平坦的范数和微分相对于在$\mathbb{R}^{n}$电流的尖锐的范数。一个时间相关的属性是自然地分配给不断发展的区域通过行动的$r$ -上链的电流与域。应用上链的表示定理的性质被证明是局部表示的$r$ -形式。利用这些概念，提出了一个广义迁移定理。
英文摘要：
A generalized transport theorem for convecting irregular domains is presented in the setting of Federer's geometric measure theory. A prototypical $r$-dimensional domain is viewed as a flat $r$-chain of finite mass in an open set of an $n$-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to be in accordance to a bi-Lipschitz type map. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in $\mathbb{R}^{n}$. A time dependent property is naturally assigned to the evolving region via the action of an $r$-cochain on the current associated with the domain. Applying a representation theorem for cochains the properties are shown to be locally represented by an $r$-form. Using these notions a generalized transport theorem is presented.
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