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 $$(m-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -chain B and a smooth compactly supported differential $$(m-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -form $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> . We are interested in minimizing $$ \textbf{M}(T) - \int _{\partial T} \Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:mi>Φ</mml:mi> </mml:mrow> </mml:math> over all m -dimensional rectifiable currents T in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> such that $$\partial T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> 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 $$\textbf{M}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> 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.