We consider homogenization of Dirichlet problem for divergence type elliptic operator when the operator is fixed, and the boundary data is oscillating, which is a particular case of the setting introduced in [2]. ...

In a bounded domain $D\subset \mathbb{R}^d$ ($d\geq 2$) consider homogenization of Dirichlet problem of the elliptic system \begin{equation} \begin{cases} -\nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u(x) = 0, & x \in D, \\ u(x) = g \left(x , \frac{x}{\varepsilon} \right), & x \in \partial D \end{cases} \tag{1} \end{equation} where $\varepsilon > 0$ is a small parameter and $A= A^{\alpha \beta } (x) \in M_N(\mathbb{R})$, $x\in \mathbb{R}^d$ is a family of functions indexed by $1\leq \alpha, \beta \leq d$ with values in the set of matrices $M_N( \mathbb{R})$. ...