  We consider a Dirichlet problem for the Allen-Cahn equation in a smooth,
bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ 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.
