A Highly Anisotropic Nonlinear Elasticity Model for Vesicles II: Derivation of the Thin Bilayer Bending Theory

Abstract

We study the thin-shell limit of the nonlinear elasticity model for vesicles introduced in part I. We consider vesicles of width \({2 \varepsilon \downarrow 0}\) with elastic energy of order \({\varepsilon^3}\). In this regime, we show that the limit model is a bending theory for generalized hypersurfaces—namely, co-dimension one oriented varifolds without boundary. Up to a positive factor, the limit functional is the Willmore energy. In the language of \({{\it \Gamma}}\)-convergence, we establish a compactness result, a lower bound result and the matching upper bound in the smooth case.