Degenerate or singular parabolic systems with partially DMO coefficients: the Dirichlet problem

Abstract

In this paper, we study solutions $u$ of parabolic systems in divergence form with zero Dirichlet boundary conditions in the upper-half cylinder $Q_1^{+}\subset {\mathbb{R}}^{n+1}$, where the coefficients are weighted by $x_n^{\hat{I}}\pm$, $\hat{I}\pm \in (-\infty ,1)$. We establish higher-order boundary Schauder type estimates of $x_n^{\hat{I}}\pm u$ under the assumption that the coefficients have partially Dini mean oscillation. As an application, we also achieve higher-order boundary Harnack principles for degenerate or singular equations with Hölder continuous coefficients.

36 pages