On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in ${\mathbf{R}}^2$

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\mathrm{\Omega }\subset {\mathbf{R}}^n\mathrm{.}$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension n=2 an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.