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.