  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|^b near s=1, with
1<b<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.
