We suppose that $\Omega \subset \mathbb{R}^2$ is an open, bounded region of a smooth boundary $\Gamma=\partial{\Omega}$, $f \in L^2(\Omega), k \in L^{\infty}(\Omega)$ and there is an $a >0$ such that $k(x) \geq a, \forall x \in \Omega$.
We want to find a solution of the elliptic problem

$\left\{\begin{matrix}
k(x)\Delta u(x)=f(x) & \text{ in } \Omega\\
u=0 & \text{in } \Gamma
\end{matrix}\right.$

We consider that the region $\Omega$ is seperated into two subregions $\Omega=\Omega_1 \cup \Omega_2$ and the function $k(x)$ is constant in each region