Mathematics Colloquia and Seminars

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Classically, Plateau's problem asks to find a surface of least area with a given boundary B. In this talk, 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 chain B and a smooth compactly supported differential form w. We are interested in minimizing M(T) - bd(T)(w) over all m-dimensional rectifiable currents T in Euclidean space such that the boundary of 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 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.