A^\epsilon_{ij}(\bm{x})\right]}_{1\leq i, j \leq d}=A(\bm{x}/\epsilon)={\left[ A_{ij}(\frac{\bm{x}}{\epsilon})\right]}_{1\leq i, j \leq d}$$ serves as the coefficients in our PDE model. The scale parameter $\epsilon \ll 1$. Also provide $f \in V^*$, $g \in L^2(\Gamma_N)$, and $j: V_j \rightarrow {\mathbb{R}}$ is a locally Lipschitz function. Now we can formulate our contact problem, $$\label{eq:contact problem}
\left\{
\begin{aligned}
-{\mathrm{div}}(A^\epsilon(\bm{x})\nabla u_\epsilon) = f ~~~~ &\text{in} ~~ \Omega \\
u_\epsilon = 0 ~~~~ &\text{on} ~~ \Gamma_D \\
\bm{n} \cdot A^\epsilon \nabla u_\epsilon = g ~~~~ &\text{on} ~~ \Gamma_N \\
-\bm{n} \cdot A^\epsilon \nabla u_\epsilon \in \partial j(\gamma_j u_\epsilon) ~~~~ &\text{on} ~~ \Gamma_C
\end{aligned}   
\right. ,$$ and its hemivariational form: $$\label{eq:contact problem hemiform}
\left\{
\begin{aligned}
&\text{Find } u_\epsilon \in V, \text{ s.t. } \forall v \in V \\
& \int_\Omega A^\epsilon \nabla u_\epsilon \cdot \nabla v + j^0(\gamma_j u_\epsilon; \gamma_j v) \geq {\left\langle f,v \right\rangle}+\int_{\Gamma_N} gv 
\end{aligned}
\right. .$$ We mention that $v \mapsto \int_{\Gamma_N} gv$ is a bounded functional on $V$ since $g \in L^2(\Gamma_N)$, one can rewrite ${\left\langle \tilde{f},v \right\rangle}\coloneqq {\left\langle f,v \right\rangle}+\int_{\Gamma_N} gv$. It is clear that ${\left\lVert \tilde{f} \right\rVert}_{V^*} \leq {\left\lVert f \right\rVert}_{V^*}+c_k {\left\lVert g \right\rVert}_{0,\Gamma_N}$, where $c_k$ equals the trace operator norm from $V$ to $L^2(\Gamma_N)$.

To make this hemivariational form solvable, we 