I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$ in using the Maximin Principle (Intuitive understanding of Maximin Principle). Is there anyone could explain this part in using $[a,b] \subset [a',b']$? I know that the eigenvalues related to these interval is one the form $\lambda_n = \frac{n^2 \pi^2}{l^2}$ ($l$ is the lenght of an interval) with the eigenfunction $u_n= \sin (\frac{n \pi}{l} x)$.
The maximin is just defined in the theorem $2$ page $324$.
Part of the proof : In the Dirichlet case, consider the maximin expression ($13$) for $D$. If $w(x)$ is any trial function in $D$, we define in all of $D'$ by setting it equal to zero outside $D$; that is,
$w'(x) = \begin{cases} w(x) & \quad \text{if } x \in D\\ 0 & \quad \text{if } x \in D' - D\\ \end{cases}. $
Thus every trial function in $D$ corresponds to a trial function in $D$ (but not conversely). So, compared to the trial functions for $D'$, the trial functions for $D$ have the extra constraint of vanishing in the rest of $D'$. By the principal ($17$), the maximin for $D$ is larger than the maximin for $D'$. It follows that $\lambda_n \geq \lambda_n'$, as we wanted to prove.