Let $U \subset \mathbb{R}^n$ be open, let $1 \leq m \leq n$, let $\Gamma \in \mathcal{I}_{m-1}(U)$ be an integral $(m-1)$-current satisfying $\partial \Gamma = 0$, and let $K \subset U$ be compact. Define
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\begin{align*}
\mathcal{A}_{\Gamma,K}(U)
:=
\{R \in \mathcal{I}_m(U) : \partial R = \Gamma,\ \operatorname{spt} R \subset K\}.
\end{align*}
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Assume that $\mathcal{A}_{\Gamma,K}(U)$ is nonempty. Then there exists $T \in \mathcal{A}_{\Gamma,K}(U)$ such that
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\begin{align*}
\mathsf{M}(T)
=
\inf\{\mathsf{M}(R) : R \in \mathcal{A}_{\Gamma,K}(U)\}.
\end{align*}