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$\dot y(t)=f(y(t),u(t))$, $y(t)\in\mathbb R^n$, $u(t)\in U\subset \mathbb R^m$.

However, suitable assumptions are needed relating $f$
with the running and exit costs.

The semiconcavity property is then applied to obtain
necessary optimality conditions,
through the formulation of a suitable version of the
Maximum Principle, and
to study the singular set of the value function.

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

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