Eigenfunctions of elliptic equations

Question

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 $\Omega$?

Answer 1

In general the answer is "no." Let $\Omega = (-\pi/2,\,\pi/2) \subset \mathbb{R}$, and for $\varphi \in C^{\infty}_0(\Omega)$ take $$u_1 = \cos(x), \quad u_2 = \cos(x) + \epsilon \varphi$$ $$a_1 = 1, \quad a_2 = \frac{\cos(x) + \epsilon \varphi}{\cos(x) - \epsilon \varphi''},$$ $$\lambda = 1.$$ For $\epsilon > 0$ small, the desired conditions are satisfied, but $a_1 \neq a_2$.