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.Comment: 27 page

A transport theorem for nonconvecting open sets on an embedded manifold

“…The transport theorems mentioned so far hold for evolving smooth domains, but results for irregular domains have been obtained. Falach and Segev established generalized transport theorems by modeling the domain of integration either as a de Rham current [9] or a flat chain [10] in the spirit of Federer's geometric measure theory [11]. In both cases, the domain was convecting according to a given flow map, though in [10] this map was only required to be Lipschitz.…”

“…Falach and Segev established generalized transport theorems by modeling the domain of integration either as a de Rham current [9] or a flat chain [10] in the spirit of Federer's geometric measure theory [11]. In both cases, the domain was convecting according to a given flow map, though in [10] this map was only required to be Lipschitz. Seguin and Fried [20] proved a transport theorem involving irregular domains using Harrison's theory [15] of differential chains.…”