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 \( \textbf{M}(T) - \int _{\partial T} \Phi \) 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 \(\textbf{M}\) 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.
