Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

Source abstract

This paper is concerned with sharp boundary regularity estimates in homogenization of the Dirichlet problem for Stokes systems. We obtain Lipschitz estimates for the velocity and $L^\infty$ estimate for the pressure, under some reasonable smoothness assumption on rapidly oscillating periodic coefficients. We find a new way to obtain the sharp uniform boundary estimates without imposing the symmetry assumption on coefficients. Additionally, we emphasize that the $L^\infty$ estimate for the pressure does require the $O(\varepsilon^{1/2})$ convergence rate, locally at least, compared to $O(\varepsilon^\lambda)$ for the velocity with $\lambda\in(0,1/2)$.