Upper Bound on the Coarsening Rate for an Epitaxial Growth Model

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.