Covers, soap films and BV functions

Giovanni Bellettini,Giuseppe Scianna,Franco Pasquarelli,Maurizio Paolini

doi:10.1515/geofl-2018-0005

Abstract

Abstract In this paper we review the double covers method with constrained BV functions for solving the classical Plateau’s problem. Next, we carefully analyze some interesting examples of soap films compatible with covers of degree larger than two: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, a soap film that retracts to its boundary, and various soap films spanning an octahedral frame.

Highlights

The method does not impose any topological restriction on the solutions; it relies on the theory of currents and takes into account unoriented objects
A slightly di erent approach has been proposed in [2]; it is based on the minimization of the total variation for functions de ned on a single covering space and satisfying a suitable constraint on the bers
Chosing the right de nition of cover depends on the structure of the minimizing solution that is desired, like e.g. the type of singularities that are allowed or orientability of the minimizing lm. It takes advantage of the full machinery known on the space of BV functions de ned on a locally Euclidean manifold: for instance, and remarkably, it allows approximating the considered class of Plateau type problems by Γ-convergence

Summary

Introduction

The method does not impose any topological restriction on the solutions; it relies on the theory of currents and takes into account unoriented objects It consists essentially in the construction of a pair of covering spaces, and is based on the minimization of what the author called the soap lm mass. A slightly di erent approach has been proposed in [2]; it is based on the minimization of the total variation for functions de ned on a single covering space and satisfying a suitable constraint on the bers This method does not impose any a priori topological restriction on the solutions.

The functional F is then de ned by

By the properties of BV functions we have

Hence uis constant on π

Covers of degree larger than two