Partial Plateau's Problem with $H$-mass

Abstract

Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially span $B$. Our boundary data is given by a flat $(m-1)$-chain $B$ and a smooth compactly supported differential $(m-1)$-form $\Phi$. We are interested in minimizing $ \mathbf{M}(T) - \int_{\partial T} \Phi $ over all $m$-dimensional rectifiable currents $T$ in $\mathbb{R}^n$ such that $\partial T$ is a subcurrent of the given boundary $B$. The existence of a rectifiable minimizer is proven with Federer and Fleming's compactness theorem. We generalize this problem by replacing the mass $\mathbf{M}$ with the $H$-mass of rectifiable currents. By minimizing over a larger class of objects, called scans with boundary, and by defining their $H$-mass as a type of lower-semicontinuous envelope over the $H$-mass of rectifiable currents, we prove an existence result for this problem by using Hardt and De Pauw's BV compactness theorem.