Sala Seminari (Dip. Matematica).
Biomembranes are remarkable structures with both fluid-like and solid-like properties: the main constituents are amphiphilic lipids, which have a head part that attracts water and a tail part that repels it. Because of these properties, such lipids organize themselves in micelle and bilayer structures, where the head parts shield the lipid tails from the contact with water. In a recent paper by Peletier and Röger (ARMA, 2009) a mesoscale model was introduced in the form of an energy for idealized and rescaled head and tails densities: the energy has two contributions, one penalizes the proximity of tail to polar (head or water) particles and the second implements the head-tail connection as an energetic penalization. The tickness of the structure is very small, and a full Gamma-convergence result has been proved in the same paper in the two-dimensional case: the Gamma-limit turns out to be the Euler elasitca functional for curves in the plane. The three-dimensional case is much harder and we have only partial results. In this seminar I will present the mesoscopic model proposed by Peletier and Röger, I will briefly explain how the deduction of the 2D-macroscopic model by Gamma-convergence works and then I will give some details on the 3D-case: the analysis of such a case requires deep tools from geometric measure theory, like currents and varifolds, in order to have weak notions of surfaces good for Calculus of Variations and for which a suitable notion of curvatures exists. The research project is in collaboration with Mark Peletier and Matthias Röger.