On the Dirichlet problem in cylindrical domains for evolution Oleinik-Radkevic PDE's: A Tikhonov-type theorem

[摘要] We consider the linear second order PDO's L = L-0 - partial derivative(t) := Sigma(N)(i,j=1) partial derivative(xi) (a(ij)partial derivative(xj)) - Sigma(N)(j=1) b(j)partial derivative(xj) - partial derivative(t) , and assume that L-0 has nonnegative characteristic form and satisfies the Oleinik-Radkevic rank hypoel-lipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for L and L-0 on bounded open subsets of RN+1 and of R-N, respectively. Our main result is the following Tikhonov-type theorem: Let O := Omega x]0, T[ be a bounded cylindrical domain of RN + 1, Omega subset of R-N, x(0) is an element of partial derivative Omega and 0 < t(0) < T. Then z(0) = (x(0), t(0)) is an element of partial derivative O is L-regular for O if and only if x(0) is L-0-regular for Omega. As an application, we derive a boundary regularity criterion for degenerate Ornstein-Uhlenbeck operators. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Dirichlet problem;Perron-Wiener solution;Boundary behavior of Perron-Wiener solutions;Exterior cone criterion;Hypoelliptic operators;Potential theory