Homogenization for a periodic elliptic operator in a strip with various boundary conditions

Abstract

A homogenization problem is considered for the periodic elliptic differential operators on $L_2(\Pi )$, $\Pi =\mathbb {R} \times (0, a)$, defined by the differential expression \begin{align*} \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D}_j g_j(x_1/\varepsilon , x_2)\mathrm {D}_j + \sum _{j=1}^2 \bigl ( h_{j}^{*}(x_1/\varepsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )& \\ + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon , x_2)& \end{align*} with periodic, Neumann, or Dirichlet boundary conditions. The coefficients of the expression are assumed to be periodic of period $1$ in the first variable and smooth in some sense in the second. Sharp-order approximations are obtained for the inverse of $\mathcal {B}_{\lambda }^{\varepsilon }$ with respect to $\mathbf {B}\bigl ( L_2(\Pi )\bigr )$- and $\mathbf {B}\bigl ( L_2(\Pi ), H^1(\Pi ) \bigr )$-norms in the small $\varepsilon$ limit with error terms of order $\varepsilon$.