Uniform boundary estimates for Neumann problems in parabolic homogenization
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.
scientific article; zbMATH DE number 7955225
scientific article