Calculus of Variations and Geometric Measure Theory

L. Lussardi - M. Röger

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

created by lussardi on 12 Jan 2015
modified on 14 Jan 2016


Published Paper

Inserted: 12 jan 2015
Last Updated: 14 jan 2016

Journal: Arch. Rational Mech. Anal.
Volume: 219
Number: 3
Pages: 985-1016
Year: 2016
Doi: 10.1007/s00205-015-0914-6


We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in R3 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 Röger (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.

Keywords: Curvature functionals, currents, generalized Gauss graphs, varifolds.