Minimizing Configurations for Elastic Surface Energies with Elastic Boundaries

TL;DR

This paper investigates critical surfaces minimizing a combined energy involving mean curvature deviations and boundary elastic energy, characterizing existence and minimizers for specific topologies and curvature conditions.

Contribution

It provides new characterizations of equilibrium surfaces with prescribed curvature and boundary elasticity, including existence criteria and explicit minimizers for annuli and discs.

Findings

Finite energy infimum conditions for topological annuli when c_o ≥ 0

Explicit minimizers identified for certain topologies

Existence results for equilibrium surfaces with prescribed curvature

Abstract

We study critical surfaces for a surface energy which contains the squared $L^2$ norm of the difference of the mean curvature $H$ and the spontaneous curvature $c_o$, coupled to the elastic energy of the boundary curve. We investigate the existence of equilibria with $H\equiv -c_o$. When $c_o \ge 0$ we characterize those cases where the infimum of this energy is finite for topological annuli and we find the minimizer in the cases that it exists. Results for topological discs are also given.