We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain Ω ⊂ ℝn. 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.
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