On the Dirichlet problem in cylindrical domains for evolution Ole\vınik--Radkevič PDE's: a Tikhonov-type theorem

Abstract

We consider the linear second order PDO's $$\mathcal{L}={\mathcal{L}}_0-{\partial }_t:=\sum _{i,j=1}^N{\partial }_{x_i}(a_{i,j}{\partial }_{x_j})-\sum _{j=i}^N{b}_j{\partial }_{x_j}-{\partial }_t,$$and assume that ${\mathcal{L}}_0$ has nonnegative characteristic form and satisfies the Ole\vınik--Radkevič rank hypoellipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for $\mathcal{L}$ and ${\mathcal{L}}_0$ 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 $\mathcal{L}$-regular for $\mathcal{O}$ if and only if $x_0$ is ${\mathcal{L}}_0$-regular for $\Omega$. As an application, we derive a boundary regularity criterion for degenerate Ornstein--Uhlenbeck operators.