\rightharpoonup \bm{p}_0 ~~~~ \bm{q}_\epsilon \rightharpoonup \bm{q}_0 ~~~~ \text{in } L^2(\Omega)^d .$$ If, in addition $\bm{q}_\epsilon = \nabla v_\epsilon$ and ${\mathrm{div}}\bm{p}_\epsilon \rightarrow f$ in $H^{-1}(\Omega)$, then $\bm{p}_\epsilon \cdot \bm{q}_\epsilon \overset{*}{\rightharpoonup} \bm{p}_0 \cdot \bm{q}_0$. Here “$\overset{*}{\rightharpoonup}$” stands for *\*-weak* convergence, it means that $\forall \phi \in C^\infty_0(\Omega), \int_\Omega \bm{p}_\epsilon \cdot \bm{q}_\epsilon \phi \rightarrow \int_{\Omega} \bm{p}_0 \cdot \bm{q}_0 \phi$.

Now we can prove the main homogenization result:

Under the assumptions \[ass:A\]-\[ass:C\], let $u_\epsilon$ and $u_0$ be the unique solution of \[eq:contact problem hemiform\] and \[eq:contact problem homohemiform\] respectively. Then $u_\epsilon \rightharpoonup u_0$ in $V$.

First, we make some justifications. Take $v \in C^\infty_0(\Omega)$ and $-v$ into \[eq:contact problem homohemiform\], it is obvious that $j^0(\gamma_ju_\epsilon; \gamma_j v)=j^0(\gamma_j u_\epsilon; -\gamma_j v)=0$. We get $\int_\Omega A^\epsilon \nabla u_\epsilon \cdot \nabla v = {\left\langle f,v \right\rangle}$, which means $-{\mathrm{div}}(A^\epsilon \nabla u_\epsilon) = f$ in the sense of weak derivative, and $f$ can naturally embed into $H^{-1}(\Omega)$.

We have already obtained that $\{ u_\epsilon\}$ is uniformly bounded, then up to a subsequence, we have $\nabla u_\epsilon \rightharpoonup \nabla u_0$, and $A^\epsilon \nabla u_\epsilon \rightharpoonup \bm{p}_0$ in $L^2(\Omega)^d$. And then we show $\bm{p}_0=\hat{A}\nabla u_0$ and $u_0$ satisfies \[eq:contact problem homohemiform\]. Combine div-curl lemma and \[thm:oscillating converge\] we get $$A^\epsilon_{ij}\partial_j 