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 $$(m-1)$$ ( m - 1 ) -chain B and a smooth compactly supported differential $$(m-1)$$ ( m - 1 ) -form $$\Phi $$ Φ . We are interested in minimizing $$ \textbf{M}(T) - \int _{\partial T} \Phi $$ M ( T ) - ∫ ∂ T Φ over all m-dimensional rectifiable currents T in $$\mathbb {R}^n$$ R n such that $$\partial T$$ ∂ 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 $$\textbf{M}$$ 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.