Uniform boundary estimates for Neumann problems in parabolic homogenization

For a family of second-order parabolic systems with periodic, oscillating and time-dependent coefficients, we establish the uniform boundary Holder and Lipschitz estimates for Neumann problems in C-1 and C-1,C-eta cylinders, respectively, by using the convergence rate method. Moreover, we establish the uniform W-1,W-p estimates for initial-Neumann problems in C-1 cylinders for 1 < p < infinity by using the real-variable method. As a by-product, we also obtain the Gaussian estimates for Neumann function and its derivatives.