Higher order parabolic systems with piecewise DMO and H\"{o}lder continuous coefficients

TL;DR

This paper establishes new regularity estimates for higher-order parabolic systems with piecewise smooth coefficients, advancing understanding of solutions in complex domains with discontinuities.

Contribution

It provides the first known $L_$ and Schauder estimates for higher-order systems with piecewise Dini mean oscillation and H"older continuous coefficients.

Findings

Established $L_$ estimates for solutions.

Proved Schauder estimates under piecewise regularity conditions.

Results are novel for higher-order elliptic and parabolic systems.

Abstract

In this paper, we are concerned with divergence form, higher-order parabolic systems in a cylindrical domain with a finite number of subdomains. We establish $L_\infty$ and Schauder estimates of solutions when the leading coefficients and the non-homogeneous term exhibit piecewise Dini mean oscillation and piecewise H\"{o}lder continuity, respectively. To the best of our knowledge, our results are new for higher-order elliptic and parabolic systems.