  We capture optimal decay for the Mullins-Sekerka evolution, a nonlocal,
parabolic free boundary problem from materials science. Our main result
establishes convergence of BV solutions to the planar profile in the physically
relevant case of ambient space dimension three. Far from assuming small or
well-prepared initial data, we allow for initial configurations that do not
have graph structure and in which the region of positive phase is not simply
connected, hence explicitly including the regime of Ostwald ripening. In terms
only of initially finite (not small) excess mass and surface energy, we
establish that the surface becomes a Lipschitz graph within a fixed timescale
(quantitatively estimated) and remains trapped within this setting for future
times. To obtain the graph structure, we leverage regularity results from
geometric measure theory. At the same time, we extend a duality method
previously employed for one dimensional PDE problems to higher dimensional,
nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy,
dissipation, and graph height are obtained.

Analysis of PDEs