  We consider a well known model for lipid-bilayer membrane vesicles exhibiting
phase separation, incorporating a phase field with finite curvature elasticity.
We prove the existence of a plethora of equilibria, corresponding to
symmetry-breaking solutions of the Euler-Lagrange equations, via global
bifurcation from the spherical state. To the best of our knowledge, this
constitutes the first rigorous existence results for this class of problems. We
overcome several difficulties in carrying this out. Due to inherent in-plane
fluidity combined with finite curvature elasticity, neither the Eulerian
(spatial) nor the Lagrangian (material) description of the model lends itself
well to analysis. Instead we adopt a singularity-free radial-map description
that effectively eliminates the grossly underdetermined in-plane fluid
deformation. The resulting governing equations comprise a quasi-linear elliptic
system with lower-order nonlinear constraints. We then show the equivalence of
our problem to that of finding the zeros of compact vector field. The latter is
not routine; we obtain certain spectral estimates and then shift the principle
part of the operator. With this in hand, we combine well known group-theoretic
ideas for symmetry-breaking with global bifurcation theory to obtain our
results.
