Asymptotic Expansion of a Non-local Isoperimetric Energy and Application to Mechanochemical Models related to Pattern Formation in Biological Membranes

We derive a macroscopic limit for a sharp interface version of a model proposed by Komura, Shimokawa and Andelman (2006) to investigate pattern formation due to competition of chemical and mechanical forces. The problem is reformulated as a non-local isoperimetric problem, for which we identify sub- and supercritical parameter regimes in terms of the relative strength of the competing local and non-local interactions. Using the Autocorrelation Function, we find an asymptotic expansion of the energy in terms of the length scale parameter up to first order and derive the Γ–limit in the subcritical regime, and the Γ–limit of the rescaled energy in the critical regime. Concerning the analysis of the Autocorrelation Function, we show that regularity near the origin is inherited from the regularity of the corresponding set and present formulas for higher derivatives at the origin.