Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. Suppose there exists $\lambda>0$ and $u_i\in H^1_0(\Omega)$ such that

$$-a_i \Delta u_i=\lambda u_i,$$ and $\frac{\partial u_1}{\partial \nu}=\frac{\partial u_2}{\partial \nu}$ on $\partial \Omega$, where $\nu$ is the unit normal vector on $\partial \Omega$. Is $a_1=a_2$ in $\Omega$?