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

We consider the linear second order PDO's $$ mathscr{L} = mathscr{L}_0 - partial_t : = sum_{i,j =1}^N partial_{x_i}(a_{ij} partial_{x_j} ) - sum_{j=i}^N b_j partial_{x_j} - partial _t,$$and assume that $mathscr{L}_0$ has nonnegative characteristic form and satisfies the Ole{i}nik--Radkevi{c} rank hypoellipticity condition. These hypotheses allow 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 $mathcal{O}:= Omega imes ]0, T[$ be a bounded cylindrical domain of $mathbb R^{N+1}$, $Omega subset mathbb R^{N},$ $x_0 in partial Omega$ and mbox{$0 < t_0 < T$}. 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 degenerate Ornstein--Uhlenbeck operators.