Snapping elastic curves as a one-dimensional analogue of two-component lipid bilayers

TL;DR

This paper introduces a mathematical model for elastic curves with two phases, mimicking lipid bilayer membranes, and demonstrates convergence of energies with numerical illustrations.

Contribution

It develops a new energy model for phase-separated elastic curves and proves their $ ext{Gamma}$-convergence to kinked curves, advancing the understanding of membrane analogues.

Findings

Energy functionals converge to kinked curve energies

Numerical examples illustrate theoretical results

Model captures phase boundary effects on curvature

Abstract

In order to study a one-dimensional analogue of the spontaneous curvature model for two-component lipid bilayer membranes we consider planar curves that are made of a material with two phases. Each phase induces a preferred curvature to the curve, and these curvatures as well as phase boundaries may lead to the development of kinks. We introduce a family of energies for smooth curves and phase fields, and we show that these energies $\Gamma$-converge to an energy for curves with a finite number of kinks. The theoretical result is illustrated by some numerical examples.