  We consider the evolution of triple junction clusters driven by the surface
diffusion flow. On the triple line we use the boundary conditions derived by
Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with
degenerated mobility. These conditions are the concurrency of the triple
junction, angle conditions between the hypersurfaces, continuity of the
chemical potentials and a flux-balance. For this system we show stability of
its energy minimizers, i.e., standard double bubbles.The main argument relies
on a Lojasiewicz-Simon gradient inequality. The proof of it differs from others
works due to the fully non-linear boundary conditions and problems with the
(non-local) tangential part.
