# Science at the triple point between mathematics, mechanics and materials science

## Publication 74

### Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional

##### Authors:

Gianni Dal Maso
SISSA,
Trieste, Italy

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Giovanni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

##### 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.
##### Get the paper in its entirety
14-CNA-019.pdf

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