Equilibrium distributions of multicomponent systems minimize the free energy
functional under the constraint of mass conservation of the components.
However, since the free energy is not convex in general,
usually one tries to characterize and to construct equilibrium distributions
as steady states of an adequate evolution equation, for example,
the nonlocal Cahn-Hilliard equation for binary alloys.
In this work a direct descent method for nonconvex functionals is
established and applied to phase separation problems in multicomponent
systems and image segmentation.