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Homogenization of the Neumann problem for higher order elliptic equations with periodic coefficients. (English) Zbl 1415.35028

This article continues the study already undertaken in other papers, by the same and also by other authors.
The goal of the paper is to give some estimates for operators \[ A_{\varepsilon} := b(D)^{\ast} g (x / \varepsilon) \, b(D), \] where \(\varepsilon\) is a positive parameter, \(g\) is a bounded and uniformly positive definite \(m \times m\)-matrix-valued periodic function, and \(b(D)\) is an operator of order \(p \geqslant 2\) of the form \[ \sum_{| \alpha | = p} b_{\alpha} D^{\alpha} \, , \qquad b_{\alpha} \ m \times n \, \text{-matrix}, \] so that \(A_{\varepsilon}\) is of order \(2 p\). The author studies the limit behaviour, as \(\varepsilon\) goes to zero, of \(A_{\varepsilon}\) in a bounded domain \(O\) of class \(C^{2p}\) under Neumann conditions. In particular, denoted by \(A^0\) the effective operator given by the expression \(b(D)^{\ast} \, g^0 \, b(D)\) and fixed \(\zeta = | \zeta | e^{i \varphi} \in \mathbb{C} \setminus \mathbb{R}_+\) with \(| \zeta | \geqslant 1\), the author proves \[ \begin{aligned} & \| ( A_{\varepsilon} - \zeta I)^{-1} - ( A^0 - \zeta I)^{-1} \|_{L^2(O) \rightarrow L^2(O)} \leqslant c \, |\zeta |^{-1+1/2p} \, \varepsilon \\ & \| ( A_{\varepsilon} - \zeta I)^{-1} - ( A^0 - \zeta I)^{-1} - \varepsilon^p K (\zeta, \varepsilon) \|_{L^2(O) \rightarrow H^p (O)} \leqslant c \, \big( |\zeta |^{-1/2+1/4p} \, \varepsilon^{1/2} + \varepsilon^p \big) \end{aligned} \] where \(K\) is the corrector which satisfies \(\| K (\zeta, \varepsilon) \|_{L^2 (O) \rightarrow H^p (O)} = O (\varepsilon^{-p})\). Due to the fact that the author considers a bounded domain it is necessary to consider also a correction term at the boundary.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J58 Boundary value problems for higher-order elliptic systems
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