Uniform boundary estimates for Neumann problems in parabolic homogenization (Q6649920)

This paper deals with the uniform boundary regularities for the classical model of parabolic systems in homogenization with Neumann boundary condition\N\[\N\begin{cases} \partial_tu_\varepsilon-\mathrm{div}(A(x/\varepsilon, t/\varepsilon^2)\nabla u_\varepsilon)=F&\text{in }\Omega\times[0, T],\\\N\frac{\partial u_\varepsilon}{\partial \nu_\varepsilon}=g &\text{on }\partial \Omega\times[0, T],\\\Nu_\varepsilon=h &\text{on } \Omega\times\{0\}, \end{cases}\N\]\Nwhere \(A(y, s)\) is \(1\)-periodic in \((y, s)\). The large-scale boundary Hölder and Lipschitz estimates in \(C^1\) and \(C^{1, \eta}\) cylinders, respectively, are established by the large-scale scheme. The uniform \(W^{1, p}\) estimates for \(1<p<\infty\) in \(C^1\) cylinders are obtained by using the real-variable method. As an application, the uniform size estimates for Neumann function as well as its derivatives are achieved.