Uniform boundary estimates in homogenization of higher-order elliptic systems

Abstract

This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $$\mathcal {L}_\varepsilon $$ , with rapidly oscillating periodic coefficients. We derive uniform boundary $$C^{m-1,\lambda } (0\!<\!\lambda \!<\!1)$$ and $$ W^{m,p}$$ 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 works “Armstrong and Smart in Ann Sci Ec Norm Super (4) 49(2):423–481 (2016) and Shen in Anal PDE 8(7):1565–1601 (2015),” is based on a suboptimal convergence rate in $$H^{m-1}(\Omega )$$ . Compared to “Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036 (2012) and Shen (2015),” 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.