Condensation of polyhedric structures onto soap films

TL;DR

This paper investigates measure-minimization problems over topological classes, establishing the existence of minimal solutions without restrictive assumptions, and linking these solutions to Almgren-minimal sets for potential applications in higher dimensions.

Contribution

It introduces a framework for existence of solutions to measure-minimization problems over broad topological classes without orientability or rectifiability constraints.

Findings

Existence of a minimal candidate as a limit of minimizing sequences.

The minimal candidate is proven to be an Almgren-minimal set.

Potential to find solutions in arbitrary dimensions using regularity results.

Abstract

We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions such as orientability or even rectifiability of surfaces. In case of problems over an open and bounded domain we establish the existence of a "minimal candidate", obtained as the limit for the local Hausdorff convergence of a minimizing sequence for which the measure is lower-semicontinuous. Although we do not give a way to control the topological constraint when taking limit yet -- except for some examples of topological classes preserving local separation or for periodic two-dimensional sets -- we prove that this candidate is an Almgren-minimal set. Thus, using regularity results such as Jean Taylor's theorem, this could be a way to find solutions to the above minimization problems under a generic setup in arbitrary dimension and codimension.