### 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 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

**Upper Bound on the Coarsening Rate for an Epitaxial Growth Model.** / Kohn, Robert; Yan, Xiaodong.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 56, no. 11, pp. 1549-1564. https://doi.org/10.1002/cpa.10103

}

TY - JOUR

T1 - Upper Bound on the Coarsening Rate for an Epitaxial Growth Model

AU - Kohn, Robert

AU - Yan, Xiaodong

PY - 2003/11

Y1 - 2003/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=1142288239&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1142288239&partnerID=8YFLogxK

U2 - 10.1002/cpa.10103

DO - 10.1002/cpa.10103

M3 - Article

VL - 56

SP - 1549

EP - 1564

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 11

ER -