Publication 14-CNA-019
**Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional**

Gianni Dal Maso

SISSA

Via Bonomea 265

34136 Trieste, Italy

Irene Fonseca

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh PA 15213-3890 USA

Giovanni Leoni

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh PA 15213-3890 USA

**Abstract: **The asymptotic behavior of an anisotropic Cahn-Hilliard functional with
prescribed mass and Dirichlet boundary condition is studied when the parameter $\epsilon$ that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to
$|s-1|^{\beta}$ near $s = 1$, with $1 < \beta < 2$. The first order term in the asymptotic
development by $\Gamma$-convergence is well-known, and is related to a suitable
anisotropic perimeter of the interface. Here it is shown that, under these
assumptions, the second order term is zero, which gives an estimate on
the rate of convergence of the minimum values.

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