  We show that certain free energy functionals that are not convex with respect
to the usual convex structure on their domain of definition, are strictly
convex in the sense of displacement convexity under a natural change of
variables. We use this to show that in certain cases, the only critical points
of these functionals are minimizers. This approach based on displacement
convexity permits us to treat multicomponent systems as well as single
component systems. The developments produce new examples of displacement convex
functionals, and, in the multi-component setting, jointly displacement convex
functionals.
