### 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.

Original language | English (US) |
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Pages (from-to) | 1549-1564 |

Number of pages | 16 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 56 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2003 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*56*(11), 1549-1564. https://doi.org/10.1002/cpa.10103