Higher regularity for solutions to equations arising from composite materials

TL;DR

This paper proves higher regularity estimates for solutions to parabolic and elliptic systems with piecewise smooth coefficients in composite materials, addressing an open problem in elliptic regularity theory.

Contribution

It establishes piecewise regularity estimates for parabolic systems with complex interfaces, extending previous results and solving an open problem in elliptic regularity.

Findings

Piecewise $C^{(s+1+ u)/2,s+1+ u}$ regularity for parabolic solutions

Estimates are independent of interface distances

Addresses an open problem in elliptic regularity theory

Abstract

We consider parabolic systems in divergence form with piecewise $C^{(s+\delta)/2,s+\delta}$ coefficients and data in a bounded domain consisting of a finite number of cylindrical subdomains with interfacial boundaries in $C^{s+1+\mu}$, where $s\in\mathbb N$, $\delta\in (1/2,1)$, and $\mu\in (0,1]$. We establish piecewise $C^{(s+1+\mu')/2,s+1+\mu'}$ estimates for weak solutions to such parabolic systems, where $\mu'=\min\big\{1/2,\mu\big\}$, and the estimates are independent of the distance between the interfaces. In the elliptic setting, our results answer an open problem (c) in Li and Vogelius (Arch. Rational Mech. Anal. 153 (2000), 91--151).