  We study a model for lipid-bilayer membrane vesicles exhibiting phase
separation, incorporating a phase field together with membrane fluidity and
bending elasticity. We prove the existence of a plethora of equilibria in the
large, 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 surface fluidity combined with finite curvature elasticity, neither
the Eulerian (spatial) nor the Lagrangian (material) description of the model
lends itself well to analysis. This is resolved via a singularity-free
radial-map description, which effectively eliminates the grossly
underdetermined in-plane deformation. The resulting governing equations
comprise a quasi-linear elliptic system with nonlinear constraints. We show the
equivalence of the problem to that of finding the zeros of compact vector
field. The latter is not routine. Using spectral and a-priori estimates
together with Fredholm properties, we demonstrate that the principal part of
the quasi-linear mapping defines an operator with compact resolvent. We then
combine well known group-theoretic ideas for symmetry breaking with global
bifurcation theory to obtain our results.
