$\Gamma$-limit for a sharp interface model related to pattern formation on biomembranes

TL;DR

This paper derives a macroscopic limit for a sharp interface model of pattern formation on biomembranes, linking it to an isoperimetric problem and identifying parameter regimes where fine patterns do or do not form.

Contribution

It introduces a $ ext{Gamma}$-convergence analysis for a biomembrane pattern formation model, connecting microscopic interactions to a macroscopic isoperimetric problem.

Findings

Ground state energy corresponds to the isoperimetric problem in the subcritical regime.

Identifies sub- and supercritical parameter regimes affecting pattern formation.

Shows that fine scale patterns do not form in the subcritical regime.

Abstract

We derive a macroscopic limit for a sharp interface version of a model proposed in [29] to investigate pattern formation due to competition of chemical and mechanical forces in biomembranes. We identify sub- and supercrital parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.