Upper bound on the coarsening rate for an epitaxial growth model

Abstract

Abstract We study a specific example of energy‐driven coarsening in two space dimensions. The energy is ∫|∇∇ u | 2 + (1 ‐ | ∇ u | 2 ) 2 ; the evolution is the fourth‐order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t 1/3 , and the energy per unit area decays like t ‐1/3 . We prove a weak, one‐sided version of the latter statement: The time‐averaged energy per unit area decays no faster than t ‐1/3 . Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles‐Giga energy. © 2003 Wiley Periodicals, Inc.