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We consider the problem of minimizing the Willmore energy connected surfaces with prescribed surface area which are confined to a finite container. To this end, we approximate the surface by a phase field function $u$ taking values close to +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 the Willmore energy, are well known. We address the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. In this article, we provide a proof of Gamma-convergence of our model to the sharp interface limit.

A Phase Field Model for the Optimization of the Willmore Energy in the Class of Connected Surfaces