Asymptotic analysis of phase field formulations of bending elasticity models

In this paper, we give the asymptotic analysis of sharp interface analysis of the phase field function in some phase field models for Willmore's problem and equilibrium lipid bilayer cell membrane problems. We derive the explicit expression of the asymptotic expansion of the phase.eld functions minimizing the Willmore energy. Based on the structure of the phase field functions obtained via the asymptotic analysis, we can then demonstrate the consistency of phase field models and the sharp interface models. Also some error estimates of energy and Euler number formulae are further analyzed. Some numerical experiments are performed to verify our assumptions and results. The results of this paper lead to a better understanding of the structure of the phase field functions in the phase field models for Willmore's problem and the equilibrium configurations of the lipid vesicle membranes.