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We consider the linear second order PDO's $$ \mathscr{L} = \mathscr{L}_0 -\partial_t : = \sum_{i,j =1}^N \partial_{x_i}(a_{i,j} \partial_{x_j} ) -\sum_{j=i}^N b_j \partial_{x_j} - \partial _t,$$and assume that $\mathscr{L}_0$has nonnegative characteristic form and satisfies theOle\v{\i}nik--Radkevi\v{c} rank hypoellipticity condition. These hypothesesallow the construction of Perron-Wiener solutions of the Dirichlet problems for$\mathscr{L}$ and $\mathscr{L}_0$ on bounded open subsets of $\mathbb R^{N+1}$and of $\mathbb R^{N}$, respectively.  Our main result is the following Tikhonov-type theorem:  Let $\mathcal{O}:= \Omega \times ]0, T[$ be a bounded cylindrical domain of$\mathbb R^{N+1}$, $\Omega \subset \mathbb R^{N},$ $x_0 \in \partial \Omega$and $0 < t_0 < T.$  Then $z_0 = (x_0, t_0) \in \partial \mathcal{O}$ is $\mathscr{L}$-regular for$\mathcal{O}$ if and only if $x_0$ is $\mathscr{L}_0$-regular for $\Omega$.  As an application, we derive a boundary regularity criterion for degenerateOrnstein--Uhlenbeck operators.

On the Dirichlet problem in cylindrical domains for evolution Ole\v{\i}nik--Radkevi\v{c} PDE's: a Tikhonov-type theorem