Convergence to the planar interface for a nonlocal free-boundary evolution

TL;DR

This paper proves that solutions to a nonlocal free boundary problem in three dimensions converge to a flat interface, with explicit decay rates, even from complex initial conditions, using geometric measure theory and duality methods.

Contribution

It establishes convergence to a planar interface for the Mullins-Sekerka evolution in 3D without small initial data assumptions, extending duality methods to higher dimensions.

Findings

Solutions become Lipschitz graphs within a fixed timescale.

Optimal decay rates for excess energy and surface height.

Convergence holds for complex initial interfaces, including Ostwald ripening regimes.

Abstract

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 interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. 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.