Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems

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.