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Kinks in two-phase lipid bilayer membranes

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least $C^1$ across phase boundaries. We consider the spontaneous curvature model for closed rotationally symmetric two-phase membranes without excluding tangent discontinuities at interfaces a priorily. We introduce a family of energies for smooth surfaces and phase fields for the lipid phases and derive a sharp interface limit that coincides with the $Γ$-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to tangent discontinuities. The theoretical result is illustrated by numerical examples.