Variational Analysis of Lipid Raft Formation: Energy Scaling, Gradient Flows, and Pattern Formation

The formation of lipid rafts in biomembranes is an example of pattern-formation phenomena that can be studied through minimization of energy which couples lipid concentration with membrane curvature. In this talk, we will discuss a scaling law result for an approximated two-phase Canham--Helfrich energy, for which novel interpolation inequalities involving fractional Sobolev seminorms were needed. We will then turn to the associated coupled Cahn--Hilliard system, viewed as a gradient flow of this energy, and discuss existence and uniqueness of solutions as well as numerical simulations illustrating the dynamics and the patterns that emerge. The talk is based on joint works with Janusz Ginster and Barbara Zwicknagl, as well as Patrick Knopf and Dennis Trautwein.