  Partial localization is the phenomenon of self-aggregation of mass into
high-density structures that are thin in one direction and extended in the
others. We give a detailed study of an energy functional that arises in a
simplified model for lipid bilayer membranes. We demonstrate that this
functional, defined on a class of two-dimensional spatial mass densities,
exhibits partial localization and displays the `solid-like' behavior of cell
membranes.
  Specifically, we show that density fields of moderate energy are partially
localized, i.e. resemble thin structures. Deviation from a specific uniform
thickness, creation of `ends', and the bending of such structures all carry an
energy penalty, of different orders in terms of the thickness of the structure.
  These 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 enforces both optimal thickness and non-fracture; if these
conditions are met, then the limit value is given by the classical Elastica
(bending) energy of the curve.
