Kinks in a model for two-phase lipid bilayer membranes

In the spontaneous curvature model for two-phase lipid bilayer membranes the shape of vesicles is governed by a combination of an elastic bending energy and an interface energy that penalises the size of phase boundaries. Each lipid phase induces a preferred curvature to the membrane surface, and these curvatures as well as phase boundaries may lead to the development of kinks.In a rotationally symmetric setting we introduce a family of energies for smooth surfaces and phase fields for the lipid components and study convergence to a sharp-interface limit, which depends on the choice of the bending parameters of the phase field model. We prove that, if kinks are excluded, our energies $Gamma$-converge to the commonly used sharp-interface spontaneous curvature energy with the additional assumption of $C^1$-regularity across interfaces. For a choice of parameters such that kinks may appear, we obtain a limit that coincides with the $Gamma$-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to kinks.We illustrate the theoretical result by some numerical examples.