On uniform \(H^2\)-estimates in periodic homogenization (Q2731101)

The authors consider a sequence \(u^\varepsilon\) in \(H^1 (\Omega)\) such that NEWLINE\[NEWLINE\begin{cases} A_\varepsilon u_\varepsilon= f\quad &\text{in } \Omega,\\ u_\varepsilon\to u^*\quad &\text{in }H^1 (\Omega)\text{ weak}, \end{cases} \tag{1}NEWLINE\]NEWLINE where \(A_\varepsilon= -{\partial\over\partial x_k} (a^\varepsilon_{kl} (x){\partial\over \partial x_l})\) with \(a^\varepsilon_{kl} (x)=a_{kl} ({x\over\varepsilon})\), \(x\in\mathbb{R}^N\). The goal of this work is to find necessary and sufficient conditions on \(a(y)\) for \(u^\varepsilon\) to be bounded in \(H^2_{\text{loc}} (\Omega)\) under the hypothesis that \(f\in L^2_{ \text{loc}}(\Omega)\). Here the authors introduce what they call regular materials for which, by definition, the corresponding solution of the classical periodic homogenization problem remains bounded in \(H^2_{\text{loc}} \), as was mentioned above. Moreover, they give examples of two types of such materials depending on whether the coefficients representing them belong to \(W^{1,\infty}\) or not.