First, consider the equation :
\begin{equation}\tag{1}\label{osmx} \begin{cases} \Delta u(x) = 2n, \ \ x \in\Omega \\ u(x) = \|x\|^2, \ \ x \in \partial\Omega \\ \end{cases} \end{equation}
where $\Omega\subset\mathbb{R}^n$ is bounded and regular enough. Obviously the unique solution of (\ref{osmx}) is $u(x) = \|x\|^2$, which is convex, and its convexity doesn't depend on the geometry of $\Omega$. Consider now the following equation: \begin{equation}\tag{2}\label{osmx2} \begin{cases} -\text{div}(a(x)\nabla u(x)) = 2n, \ \ x \in\Omega \\ u(x) = \|x\|^2, \ \ x \in \partial\Omega \\ \end{cases} \end{equation}
where $a$ is defined as follow: \begin{equation}\label{eq:osmx3} a(x) = \begin{cases} \ a_1, \ \ x \in\Omega_1 \\ \ a_2, \ \ x \in\Omega_2 \\ \end{cases} \end{equation}
Where $\Omega_1 \subset\Omega$ is open and $\Omega_2 = \Omega\setminus\overline\Omega_1$ and $a_1, a_2\in \mathbb{R}$. This problem is called elliptic transmission problem. I have proved that if $a_1-a_2$ is small enough, the solution of (\ref{osmx2}) restricted to each $\Omega_i$ is convex. I did that using the transmission conditions, regular estimations, Sobolev inequalities and $C^2$ proximity of the solution with the solution of (\ref{osmx}). I never used geometry hypothesis, and I want to know if that problem (generalizations) is well known, or any idea to prove convexity, maybe with more flexible conditions on $a_i$.