Global minimizers for axisymmetric multiphase membranes

We consider a Canham−Helfrich−type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham−Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase [R. Choksi and M. Veneroni, Calc. Var. Partial Differ. Equ. (2012). DOI:10.1007/s00526-012-0553-9] and prove existence of a global minimizer.