I have a few difficulties understanding the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$. I've managed to prove that $N(\lambda) \leq \frac{\lambda ab}{4 \pi}$, but not $N(\lambda) \geq \frac{\lambda ab}{4 \pi} - C \sqrt{\lambda}$, for some constant $C$. Is there anyone could give me a good hint to solve this problem?
The eigenvalues $$ \lambda_{ab} = \pi^2\bigg(\frac{1}{a^2} + \frac{1}{b^2}\bigg) $$ for $a,b \in \Bbb{N}$ are squared lengths of vectors in the lattice $(\pi/a)\mathbb{N}\times(\pi/b)\mathbb{N}$, where I let $\Bbb{N} = \{1,2,3,\ldots\}$ be the positive integers. So the counting function $N(\lambda)$ counts lattice points (in the upper quadrant) whose length is less than $\sqrt{\lambda}$.
The first trick here is to notice that lattice points are uniquely associated with rectangles of dimension $(1/a)\times(1/b)$. Each rectangular chamber in the lattice has a corner furthest from the origin. Identifying lattice point with rectangle this way is a bijection. I think of them as tiles. Now note that each rectangular tile has volume $\pi^2/(ab)$. A tile is contained in the (quarter)-circle of radius $\sqrt{\lambda}$ if and only if its lattice point is counted in $N(\lambda)$, so the total area of tiles is less than the area of the quarter-circle. After multiplying through by $ab/\pi^2$: $$N(\lambda) \leq \frac{1}{4\pi}(ab)\lambda$$ This should be more or less how you derived your upper bound.
The second trick is to observe: if we reduce the quarter-circle's radius by the diameter of a rectangular tile, it is contained inside the union of the tiles. Therefore we have a lower bound, and after multiplying through by $ab/\pi^2$ we obtain the lower bound $$ \frac{1}{4\pi}ab\bigg(\sqrt{\lambda} - \operatorname{diameter of tile}\bigg)^2 \leq N(\lambda) $$
This argument is due I believe to Gauss, also in Rayleigh-Jeans 1905, Weyl 1910, and see also Courant-Hilbert Methods of Mathematical Physics. As an exercise, can you work out the Weyl's law for Neumann eigenvalues? For $n$-dimensional parallelepipeds?