The Euler-Helfrich Functional

TL;DR

This paper studies equilibrium shapes of surfaces with energies involving mean curvature deviations and boundary elastic energy, showing axially symmetric critical surfaces satisfy a simplified variational problem and providing multiple examples.

Contribution

It introduces a simplified second order variational problem for axially symmetric critical surfaces of the Euler-Helfrich functional and presents numerous solutions.

Findings

Axially symmetric critical surfaces satisfy a reduced variational problem.

Multiple explicit solutions of the simplified problem are constructed.

The study advances understanding of surface energies involving mean curvature deviations.

Abstract

We investigate equilibrium configurations for surface energies which contain the squared $L^2$ norm of the difference of the mean curvature H and the spontaneous curvature $c_o$ coupled with the elastic energy of the boundary curve, which we studied previously in [23]. It is shown that if a critical surface for this type of functional is axially symmetric, then it satisfies a simpler second order variational problem. Many examples of solutions of this are given.