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 stabilise, and their parameter dependence is further explored.

中文翻译：

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小变形球形生物膜的相分离域形成

我们推导并分析了一种能量来模拟生物膜上的脂筏形成，涉及局部平均曲率和局部组成之间的耦合。我们应用了 Fritz、Hobbs 和第一作者最近引入的一种微扰方法，将表面的几何形状描述为未变形的 Helfrich 能量最小化表面上的图。结果是表面 Cahn-Hilliard 泛函与小变形能量相结合。我们证明了这种能量的合适的最小化器存在，并考虑了具有守恒 Allen-Cahn 动力学的梯度流，证明了存在性和唯一性结果。最后，数值模拟表明，对于长期行为，筏状结构可以出现和稳定，并进一步探讨了它们的参数依赖性。