Eigenfunctions of elliptic equations

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 $$\Omega$$?

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 $$a_1 \neq a_2$$.

• Thank you Connor. Do you think the above could also hold for more than one eigenvalue? Commented Mar 21, 2019 at 14:26