Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$.
Clearly we have $\Omega_1\subset \Omega$.
Now, let $\lambda$ be the smallest eigenvalue of operator $-\Delta$ in $\Omega$ and $\lambda_1$ be the smallest eigenvalue of operator $-\Delta$ in $\Omega_1$.
We all use Neumann boundary condition.
My question: do we necessarily have $\lambda_1\geq\lambda$?
I know in generally we should. Say, if $\Omega_1$ is a small cube and $\Omega$ is a bigger cube, then if $u_1$ is an eigenfunction in $\Omega_1$, then $u_1$ is also an eigenfunction in $\Omega$. But this does not hold in my case.