Elastic bilayer membranes under confinement -- Existence, regularity, and rigidity

TL;DR

This paper investigates the existence, regularity, and rigidity of elastic bilayer membranes confined within a container, providing mathematical analysis of minimizers of Helfrich energy and conditions for spherical solutions.

Contribution

It establishes existence of minimizers as bubble trees of spherical weak branched immersions and analyzes their regularity and rigidity properties under confinement.

Findings

Existence of minimizers in bubble tree class

Optimal regularity at branch points

Rigidity results for spherical minimizers within certain parameters

Abstract

Motivated by applications to cell biology, we study the constrained minimization of the Helfrich energy among closed surfaces confined to a container. We show existence of minimizers in the class of bubble trees of spherical weak branched immersions and derive the Euler--Lagrange equations which involve a measure-valued Lagrange multiplier that is concentrated on the coincidence set with the container boundary. We provide a careful analysis of this elliptic system and prove optimal regularity for solutions throughout the branch points. For surfaces confined in the unit ball we show that the minimization problem behaves rather rigid and identify a parameter range for which minimizers are always round spheres.