Publication 14-CNA-019
Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional
Gianni Dal Maso
SISSA
Trieste, Italy
dalmaso@sissa.it
Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu
Giovanni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
giovanni@andrew.cmu.edu
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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