Calculus of variations and optimal control; optimization Definition

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Definition

Let $U \subseteq \R^n$ be an open set and let $0 \le m \le n$ . Let $\mathcal{D}^m(U)$ be the real vector space of smooth differential $m$ -forms on $U$ with compact support in $U$ . A sequence $(\omega_k)$ in $\mathcal{D}^m(U)$ converges to $0$ if two conditions hold.

There is a compact set $K \subseteq U$ that contains the support of every $\omega_k$ .
Every partial derivative of every coefficient of $\omega_k$ converges to $0$ uniformly on $K$ .

An $m$ -dimensional current on $U$ is a linear map $T \colon \mathcal{D}^m(U) \to \R$ such that $T(\omega_k) \to 0$ whenever $\omega_k \to 0$ in this sense. The boundary of an $m$ -current $T$ with $m \ge 1$ is the $(m-1)$ -current $\partial T$ given by $\partial T(\omega) = T(d\omega)$ .

Remarks

Example: let $S \subseteq U$ be a compact oriented $m$ -dimensional $C^1$ submanifold, possibly with boundary. Then $[\![S]\!](\omega) = \int_S \omega$ is linear in $\omega$ . It is continuous, since $\abs{\int_S \omega} \le \operatorname{area}(S) \cdot \sup_S \abs{\omega}$ . So $[\![S]\!]$ is an $m$ -current, and Stokes' theorem gives $\partial [\![S]\!] = [\![\partial S]\!]$ . Non-example: the map $\omega \mapsto \abs{\int_S \omega}$ is continuous but not linear, so it is not a current. The $0$ -currents on $U$ are the distributions on $U$ . The mass of $T$ is the supremum of $T(\omega)$ over the forms $\omega$ with $\abs{\omega(x)} \le 1$ for all $x$ , for a suitable norm on $m$ -covectors.

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Differential form concept
Open set concept
Current definition

Tag	`Z6QXZX1THM`
URI	`m2://Z6QXZX1THM`
Kind	definition
Writing	W0 stub
MSC	49Q15, 58A25
Defines	Current
Uses	Differential form, Open set
Changed	2 November 2026

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