Domain formation via phase separation for spherical biomembranes with small deformations

We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn-Hilliard functional coupled with a small deformation energy We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen-Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long time behaviour raft-like structures can emerge and stablise, and their parameter dependence is further explored.