We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in 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 Roeger [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 semicontinuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds.
- Curvature functionals
- Generalized Gauss graphs