Regarding the Euler-Plateau Problem with Elastic Modulus

TL;DR

This paper investigates equilibrium shapes of elastic membranes influenced by the Euler-Plateau energy, revealing how physical parameters affect minimizers and boundary data determine critical surface areas.

Contribution

It introduces a comprehensive analysis of the Euler-Plateau problem with elastic modulus, linking boundary data to critical surface areas and characterizing symmetric solutions as planar elasticae.

Findings

Potential minimizers depend heavily on rigidity parameters.

Critical surface areas can be computed from boundary data.

Axially symmetric critical surfaces are planar elasticae.

Abstract

We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can be computed entirely from their boundary data. When the elastic modulus does not vanish, it is shown that axially symmetric critical immersions and critical immersions of disk type are necessarily planar domains bounded by area-constrained elasticae. The cases of topological genus zero with multiple boundary components and unrestricted genus with control on the geodesic torsion are also discussed, and sufficient conditions are given which establish the same conclusion in these cases.