Compare of eigenvalues $λ_{1,\Omega_2}$, $λ_{1,\Omega}$ and $λ_{1,\Omega_1}$

Question

Recall that $\lambda_1$ is the smallest eigenvalue of the Laplacian with boundary conditions of Dirichlet and we know that the link between $\lambda_1$ and the smallest possible constant in the inequality of Poincaré

$$||v||^2_{L^2(\Omega)}\leq c ||\nabla v||^2_{L^2(\Omega)},\forall v\in H^0_1(\Omega)$$

is $\lambda_1=\frac{1}{c}$.

Now, consider two bounded domains $\Omega_1$ and $\Omega_2$ of $\mathbb{R}^2$ such that $\Omega_1\subseteq \Omega\subseteq\Omega_2$.

To show that $$λ_{1,\Omega_2}\leq λ_{1,\Omega}\leq λ_{1,\Omega_1}$$

Thanks in advance for your help and ideas

Answer 1

The inclusion $\Omega_1\hookrightarrow\Omega$ induces an isometric embedding $H^1_0(\Omega_1)\hookrightarrow H^1_0(\Omega)$. The image $u$ of the first Dirichlet eigenfunction of $\Omega_1$ in $H^1_0(\Omega)$ therefore has $\|\nabla u\|_2^2/\|u\|_2^2 = \lambda_{1,\Omega_1}$ and can be used as a test function to estimate $\lambda_{1,\Omega}$ from above.