Gamma convergence of a family of surface--director bending energies with small tilt

TL;DR

This paper establishes a Gamma-convergence result for a family of surface bending energies with small tilt, relevant for modeling bilayer membranes, using geometric measure theory tools to analyze the limit behavior.

Contribution

It provides the first Gamma-convergence analysis of surface--director energies in three dimensions with small tilt, connecting to membrane models.

Findings

Gamma-liminf estimate established in 3D

Uses geometric measure theory for compactness

Connects to curvature energies in membrane models

Abstract

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in $\mathbb{R}^3$ equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such type of energies for example arise in a model for bilayer membranes introduced by Peletier and R\"oger [Arch. Ration. Mech. Anal. 193 (2009)]. Here we prove in three space dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to a specific curvature energy. In order to obtain appropriate compactness and lower semi-continuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds.