need the following assumptions:

-   The coefficient matrix $A(y)$ is symmetric and uniformly elliptic: $$\label{ass:A}
        \begin{aligned}
        & A_{ij}(\bm{y}) = A_{ji}(\bm{y}) \\
        & \kappa_1 {\left\lvert \xi \right\rvert}^2 \leq A_{ij}(\bm{y})\xi_i \xi_j \leq \kappa_2 {\left\lvert \xi \right\rvert}^2 ~~~~ \text{for a.e. } \bm{y} \in {\mathbb{R}}^d \text{ and } \xi \in {\mathbb{R}}^d
        \end{aligned} ~ .$$

-   There exist constants $c_0,c_1,\alpha_j$, such that: $$\label{ass:B1}
        {\left\lVert x^* \right\rVert}_{V_j^*} \leq c_0+c_1{\left\lVert x \right\rVert}_{V_j} \forall x \in V_j, \forall x^* \in \partial j(x) ,$$ $$\label{ass:B2}
        j^0(x_1; x_2-x_1)+j^0(x_2; x_1-x_2) \leq \alpha_j {\left\lVert x_1-x_2 \right\rVert}_{V_j}^2 \forall x_1, x_2 \in V_j .$$

-   Let $c_j={\left\lVert \gamma_j \right\rVert}_{V\rightarrow V_j}$ be the operator norm, there exists that, $$\label{ass:C}
        \Delta \coloneqq \kappa_1 - \alpha_j c_j^2 > 0 .$$

We remark here that assumptions **B** and **C** follow [@Han2017]. Then by utilizing the framework constructed in [@Han2017] following theorem is obvious:

The solution of problem \[eq:contact problem hemiform\] exists and is unique.

The assumptions in former work [@Liu2008] are sightly different with [@Han2017]. For example, [@Liu2008] needs $j^0_N(x, r; -r) \leq d_N(1+{\left\lvert r \right\rvert})$ which is redundant in our case. Hence, we think the assumptions in [@Han2017] are more reasonable.

We close this section by illustrating a specific contact problem. Here \[fig:Rubin\] describes a domain $\Omega$ 