Convergence rates for linear elasticity systems on perforated domains

Abstract

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $$L^{\frac{2d}{d-1-\tau }}$$ -error estimates $$O\big (\varepsilon ^{1-\frac{\tau }{2}}\big )$$ for all $$\tau \in (0,1)$$ in a bounded smooth domain, which is new even for homogenization problems on unperforated domains. It followed from weighted Hardy–Sobolev’s inequalities (given by Lehrback and Vahakangas in J Funct Anal 271(2):330–364, 2016) and a suboptimal error estimate for the square function of the first-order approximating corrector (earliest investigated by Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036, 2012) under additional regularity assumption on coefficient). The new approach relied on the weighted quenched Calderon–Zygmund estimate (initially appeared in Gloria et al. work Milan J Math 88(1):99–170, 2020 for a quantitative stochastic homogenization theory). The second effort was $$L^2$$ -error estimates $$O\big (\varepsilon ^{\frac{5}{6}} \ln ^{\frac{2}{3}}(1/\varepsilon )\big )$$ for a Lipschitz domain, followed from a duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem and local-type Sobolev–Poincare inequalities on perforated domains. Throughout the paper, we do not impose any smoothness assumption on the coefficients.