2006/07/12 by Mark A. Peletier, Peletier, Mark A., Matthias Roeger +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #49Q10 #49Q15 #49Q20 #74Q99 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Lipid Membrane Structure and Behavior #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:49Q10 #msc:49Q15 #msc:49Q20 #msc:74Q99
paper · pdf · doi:10.48550/arxiv.math-ph/0607024
56 pages, latex with figures included with \includegraphics and geompsfi.sty
arxiv created 2006/07/12 · openalex publication_date 2006/07/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.