|Location:||MSRI: Simons Auditorium|
Our goal is to formulate a methodology for solving PDEs for and on evolving surfaces. In the first part I will discuss the problem of minimizing a Helfrich (Willmore) type energy coupled to a surface Cahn Hilliard energy. This is a model for two phase biomembranes which form the boundary of a vesicle.The surface PDEs characterising a minimizer will be derived. Using gradient flow I will show how one may seek minimizers by looking for stationary solutions of a geometric fourth order evolution PDE for an evolving surface whose motion is coupled to an equation of Allen-Cahn type on that surface. One may view this as an example of a free boundary on a free boundary.
In the second part I will discuss numerical methods based on phase field models and surface finite elements. In particular I will describe double obstacle phase field models and the evolving surface finite element method.