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

TL;DR

This paper develops higher-order boundary estimates and Harnack principles for degenerate or singular parabolic systems with weighted coefficients, advancing understanding of boundary behavior in such complex PDEs.

Contribution

It introduces new boundary Schauder estimates and Harnack principles for degenerate or singular parabolic systems with partially Dini mean oscillation coefficients.

Findings

Established higher-order boundary Schauder estimates for weighted parabolic systems.

Proved higher-order boundary Harnack principles for degenerate or singular equations.

Extended boundary regularity theory to systems with weighted coefficients.

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 $x_n^\alpha$, $\alpha\in(-\infty,1)$. We establish higher-order boundary Schauder type estimates of $x_n^\alpha u$ 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 H\"older continuous coefficients.