Speaker
Description
In this talk, we explore a scaling law for the infimal energy of a one-dimensional nonlocal variant of the Canham-Helfrich functional in terms of problem parameters. This functional models the formation of periodic structures in cellular membranes, known as lipid rafts, by incorporating a coupling term between the order parameter and the local curvature of the membrane. A key tool in establishing the scaling result is a new set of nonlinear interpolation inequalities in fractional Sobolev spaces. Some of these inequalities, which bound the fractional Sobolev seminorm in terms of a Modica-Mortola energy, are used to obtain an Ansatz-free lower bound for the functional. On the other hand, to show the upper bound we use suitable periodic test functions, which depend on the parameter regime. Additionally, some results for the two-dimensional functional will be presented. The talk is based on joint work with Barbara Zwicknagl and Janusz Ginster.
