On the role of sharp chains in the transport theorem

Abstract

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.