Convergence rates of solutions for elliptic reiterated homogenization problems. (English) Zbl 1459.35025
The paper under review deals with the asymptotic behavior of the solutions to reiterated homogenization problems with Neumann boundary conditions. Namely, the authors consider the problem
\[
\left\{
\begin{array}{ll}
L_\varepsilon u_\varepsilon =-\frac{\partial}{\partial x_i}\Bigg(a_{ij}(x/\varepsilon,x/\varepsilon^2)\frac{\partial u_\varepsilon}{\partial x_j}\Bigg)=f & \mathrm{in}\ \Omega,\\
\frac{\partial u_\varepsilon}{\partial \nu_\varepsilon}=0 & \mathrm{on}\ \partial \Omega,
\end{array}
\right.
\]
where \(\Omega\) is a bounded \(C^{1,1}\) domain in \(\mathbb{R}^n\), \(\frac{\partial u_\varepsilon}{\partial \nu_\varepsilon}=n_ia_{ij}\frac{\partial u_\varepsilon}{\partial x_j}\) where \(n(x)\) is the outward unit normal to \(\partial \Omega\) at \(x\). The symmetric coefficient matrix \(A\) satisfies ellipticity, periodicity, and smoothness assumptions. The authors obtain convergence rates of the solution \(u_\varepsilon\) to the function \(u_0\) which solves the homogenized problem.
Reviewer: Paolo Musolino (Padova)
MSC:
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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