will give the definition of correctors and present that the problem \[eq:contact problem hemiform\] has a homogenized version, then we use div-curl lemma to prove $u_\epsilon$ can weak converge to the homogenized solution.

We denote $N_l(\bm{y})$ as correctors for $A^\epsilon$, which satisfy a group of PDEs with periodic boundary condition: $$\label{eq:correctors}
\left\{
\begin{aligned}
& -{\mathrm{div}}{\left( A(\bm{y})\nabla N_l \right)}={\mathrm{div}}(Ae_l)=\partial_i A_{il}(\bm{y}) ~~~~ \text{ in }Q\\
& N_l(\bm{y}) \in H^1_\sharp(Q) \text{ and } \int_Q N_l = 0
\end{aligned}
\right. .$$ For correctors, we have ${\left\lVert N_l \right\rVert}_{1,Q} \leq C(\kappa_1, \kappa_2)$. The homogenized coefficients are defined by $\hat{A}_{il}=\fint_Q A_{il}+A_{ij}\partial_j N_l {\mathrm{d}}\bm{y}$. The next lemma characterizes the homogenized coefficients $\hat{A}$, the proof can be found in [@Jikov1994] sect. 1.6.

Let $\kappa_1, \kappa_2$ define as previous, then $\hat{A}$ is symmetric and uniformly elliptic, means the following relation $$\begin{aligned}
    & \hat{A}_{ij} = \hat{A}_{ji} \\
    & \tilde{\kappa}_1 {\left\lvert \xi \right\rvert}^2 \leq \hat{A}_{ij}(\bm{y})\xi_i \xi_j \leq \tilde{\kappa}_2 {\left\lvert \xi \right\rvert}^2 ~~~~ \forall \xi \in {\mathbb{R}}^d
    \end{aligned}$$ holds, where $\tilde{\kappa}_1, \tilde{\kappa}_2$ depends on $\kappa_1, \kappa_2$.

Then we have the homogenized hemivariational form: $$\label{eq:contact problem homohemiform}
\left\{
\begin{aligned}
&\text{Find } u_0 \in V, \text{ s.t. } \forall v \in V \\
& \int_\Omega \hat{A} \nabla u_0 \cdot \nabla v + j^0(\gamma_j u_0; 