Schauder type estimates for degenerate or singular parabolic systems with partially DMO coefficients

TL;DR

This paper establishes boundary Schauder estimates for degenerate or singular parabolic systems with partially Dini mean oscillation coefficients, and extends higher-order boundary Harnack principles to all orders for certain parabolic equations.

Contribution

It provides new boundary Schauder estimates for systems with complex coefficients and extends higher-order boundary Harnack principles to all orders, broadening previous results.

Findings

Boundary Schauder estimates under partially Dini mean oscillation

Extension of higher-order boundary Harnack principles to all orders

Application to degenerate or singular parabolic systems

Abstract

We study elliptic and parabolic systems in divergence form with degenerate or singular coefficients. Under the conormal boundary condition on the flat boundary, we establish boundary Schauder type estimates when the coefficients have partially Dini mean oscillation. Moreover, as an application, we achieve $k^{\text{th}}$ higher-order boundary Harnack principles for uniformly parabolic equations with H\"older coefficients, extending a recent result in [Audrito-Fioravanti-Vita 25] from $k\ge2$ to any $k\ge1$.