Convergence rates for elliptic homogenization problems in Lipschitz domains

In this talk, I plan to study convergence rates in $L^2$ norm for elliptic homogenization problems in Lipschitz domains. It involves some new weighted-type inequalities for the smoothing operator at scale $\epsilon$, as well as, layer and co-layer type estimates, and the related details will be touched. In order to obtain a sharp result, a duality argument will be imposed. Here we do not require any smoothness assumption on the coefficients, and the main ideas may be extended to other models, such as Stokes systems and parabolic systems, arising in the periodic homogenization theory.