A phase field model for Willmore’s energy with topological constraint

Patrick Dondl (Freiburg, Germany)

We consider the problem of minimizing Willmore’s energy on confined and connected surfaces with prescribed surface area. To this end, we approximate the surface by a level set function u admitting the value +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of u. A diffuse interface approximation for the area functional, as well as for Willmore’s energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a penalization of a geodesic distance which is chosen to be sensitive to connected components of the phase field level sets and provide a proof of Gamma-convergence of our model to the sharp interface limit. Furthermore, we show some numerical results. This is joint work with Stephan Wojtowytsch (Durham University) and Antoine Lemenant (Universit Paris 7).

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