A direct approach to Plateau's problem in any codimension

TL;DR

This paper introduces a direct method for solving Plateau's problem in higher codimensions by minimizing Hausdorff measure among rectifiable sets, ensuring existence and partial regularity of solutions.

Contribution

It extends the direct approach to Plateau's problem to any codimension, providing existence and regularity results under broad conditions.

Findings

Existence of minimizers for the problem in higher codimension.

Minimizers are regular except on a small singular set.

The approach generalizes previous methods to more complex cases.

Abstract

This paper aims to propose a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of $d$-rectifiable closed subsets of $\mathbb R^n$: following the previous work \cite{DelGhiMag} the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than $(d-1)$.