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In many applications structures can be described as (local) minimizer of suitable bending energies. The most prominent example is the variational characterization of shapes of biomembranes by the use of Helfrich- or Willmore-type functionals. Whereas the restriction to topological spheres is natural in many applications, it is sometimes more reasonable to consider the class of orientable connected surfaces of arbitrary genus instead. For example the inner membrane of mitochondria cells shields the inside matrix from the outside but shows - in contrast to old textbook illustrations - a lot of of handle-like junctions and therefore represents a higher genus surface. In this example another natural constraint comes into play, given by the confinement of the inner membrane to a 'container' that is given by the outer membrane of the mitochondria.
This motivates to consider the following variational problem: Minimize Willmore's energy in the class of all compact, connected, orientable surfaces without boundary that are embedded in a bounded domain and have prescribed surface area.
We consider a phase field approximation to this problem, i.e., 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 $u$. Diffuse interface approximations for the area functional, as well as for the Willmore energy are well known. We address the main difficulty, namely 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. We present a proof of Gamma-convergence of our model to the sharp interface limit. This is joint work with Matthias Röger (TU Dortmund) and Luca Mugnai (MPI Leipzig).
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