Asymptotic solutions and bifurcation analysis of two-dimensional lipid vesicles

We derive asymptotic solutions of the shape equations for two-dimensional free lipid vesicles near the critical pressure, at which noncircular equilibrium shapes emerge. For the resulting noncircular branches, we determine the loss-of-convexity pressure along with obtaining asymptotic expressions for the enclosed area, bending energy and total free energy as functions of the osmotic pressure difference across the vesicle and validate these predictions against numerical solutions. We further show that the circular solution loses stability at the critical pressure through a mechanism consistent with a supercritical pitchfork bifurcation.