Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems

TL;DR

This paper establishes uniform boundary estimates for higher order elliptic systems with oscillating periodic coefficients, improving previous results by removing symmetry and regularity assumptions, and providing new convergence rate insights.

Contribution

It derives uniform boundary estimates for higher order elliptic systems without symmetry assumptions, using a novel convergence rate approach based on suboptimal convergence in $H^{m-1}( abla)$.

Findings

Established uniform boundary $C^{m-1, u}$ and $W^{m,p}$ estimates.

Achieved boundary $C^{m-1,1}$ estimates in less regular domains.

Provided convergence rates that do not require symmetry or extra regularity assumptions.

Abstract

This paper focuses on the uniform boundary estimates in homogenization of a family of higher order elliptic operators $\mathcal{L}_\epsilon$, with rapidly oscillating periodic coefficients. We derive uniform boundary $C^{m-1,\lambda} (0\!<\!\lambda\!<\!1)$, $ W^{m,p}$ estimates in $C^1$ domains, as well as uniform boundary $C^{m-1,1}$ estimate in $C^{1,\theta} (0\!<\!\theta\!<\!1)$ domains without the symmetry assumption on the operator. The proof, motivated by the profound work "S.N. Armstrong and C.~K. Smart, Ann. Sci. \'Ec. Norm. Sup\'er. (2016), Z. Shen, Anal. PDE (2017)", is based on a suboptimal convergence rate in $H^{m-1}(\Omega)$. Compared to "C.E. Kenig, F. Lin and Z. Shen, Arch. Ration. Mech. Anal. (2012), Z. Shen, Anal. PDE (2017)", the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $u_0$ (the solution to the homogenized problem), and thus might be of some independent interests even for second order elliptic systems.