We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. 
\begin{equation*}
\left\lbrace
\begin{split}
\nabla\cdot u=f_{1},\ & \text{in}\ \Omega_{1}=(-1,1)\times(0,1), \\
-\Delta v=f_{2},\ & \text{in}\ \Omega_{2}=(-1,1)\times(-1,0), \\
u\cdot n=-\nabla v\cdot n+g,\ & \text{on}\ \Gamma=(-1,1)\times\{0\}, \\
u=0,\ & \text{on}\ \Gamma_{1}=(-1,1)\times\{1\}, \\
\partial_{x_{2}}v=0,\ & \text{on}\ \Gamma_{2}=(-1,1)\times\{-1\}.
\end{split}
\right.
\end{equation*}

Clearly, the existence of solution requires a compatiablity condition
$$\int_{\Gamma}gdl=\int_{\Omega_{1}}f_{1}+\int_{\Omega_{2}}f_{2}.$$

How to solve $(u,v)$ and prove the estimate
$$||u||_{H^{1}(\Omega_{1})}+||v||_{H^{1}(\Omega_{2})}\leq C(||g||_{H^{\frac{1}{2}}(\Gamma)}+||f_{1}||_{L^{2}(\Omega_{1})}+||f_{2}||_{H^{-1}(\Omega_{2})}).$$