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 $\Omega$? 



Eigenfunctions of elliptic equations
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 $a\_1 \neq a\_2$.


