Three-dimensional Gamma-limit conjecture for membranes, tubes, and micelles

Let $F_\rho$ be the energy functional, let $m$ be the prescribed mass, let $\zeta>0$ be the mass ratio, and define

$$F'_\rho:=\frac{F_\rho-c(\zeta)m}{\rho^2}.$$

Here $\rho\to0$ is the vanishing-thickness parameter, $n=3$, and $\zeta_0<\zeta_1<\zeta_2$ are the transition values from the asymptotic energy formula.

Three-dimensional Gamma-limit conjecture. For $n=3$ and $\rho\to0$, if $\zeta\in(\zeta_0,\zeta_1)$, the Gamma-limit of $F'_\rho$ is a quadratic form in the principal curvatures for closed surfaces, in the sense of Radon measures. If $\zeta\in(\zeta_1,\zeta_2)$, it is the elastica functional for closed $W^{2,2}$ curves in $\mathbb R^3$, in the sense of Radon measures. If $\zeta\in(\zeta_2,\infty)$, it is a mass-partition functional for weighted Dirac delta point measures.

This conjecture predicts distinct curvature and concentration limits for the three-dimensional bilayer, cylindrical-micelle, and spherical-micelle regimes. The source gives analogies with existing variational conjectures and theorems, but no proof of these Gamma-limits.