We study an energy functional that arises in a simplified two-dimensional model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of spatial mass densities, favours concentrations on ‘thin structures’. Stretching, fracture and bending of such structures all carry an energy penalty. In this sense we show that the models captures essential features of lipid bilayers, namely partial localisation and a solid-like behaviour. Our findings are made precise in a Gamma-convergence result. We prove that a rescaled version of the energy functional converges in the ‘zero thickness limit’ to a functional that is defined on a class of planar curves. Finiteness of the limit value enforces both optimal thickness and non-fracture; if these conditions are met, then the value of this functional is given by the classical Elastica (bending) energy.
|Title of host publication||Proceedings 77th Annual Meeting of the Gesellschaft für Angewandte Mathematik und Mechanik e.V. (GAMM 2006, Berlin, Germany, March 27-31, 2006)|
|Publication status||Published - 2006|
|Name||PAMM, Proceedings in Applied Mathematics and Mechanics|
Röger, M., & Peletier, M. A. (2006). Cell membranes, lipid bilayers and the elastica functional. In Proceedings 77th Annual Meeting of the Gesellschaft für Angewandte Mathematik und Mechanik e.V. (GAMM 2006, Berlin, Germany, March 27-31, 2006) (pp. 11-14). (PAMM, Proceedings in Applied Mathematics and Mechanics; Vol. 6(1)). https://doi.org/10.1002/pamm.200610004