The following comes from Springer Online Reference Works:
Consider a bounded domain $\Omega\subset\mathbb{R}^n$ with a piecewise smooth boundary $\partial\Omega$. $\lambda$ is a Dirichlet eigenvalue of $\Omega$ if there exists a function $u\in C^2(\Omega)\cap C^0(\bar{\Omega})$ (a Dirichlet eigenfunction) satisfying the following Dirichlet boundary value problem
$$
-\Delta u=\lambda u \qquad \text{in } \Omega
$$
$$
u=0\qquad \text{in } \partial\Omega
$$
Provided $\Omega$ is bounded and the boundary $\partial \Omega$ is sufficiently regular, the Dirichlet Laplacian has a discrete spectrum of infinitely many positive eigenvalues with no finite accumulation point:
$$
0<\lambda_1\le\lambda_2\le\cdots
$$
The Weyl’s asymptotic law says that:
For large values of $k$ , if $\Omega \subset \mathbb{R}^n$ ,then
$$
\lambda_k\approx\frac{4\pi^2k^{2/n}}{(C_n\vert\Omega\vert)^{2/n}}
$$
where $\vert\Omega\vert$ and $C_n$ are the volumes of $\Omega$ and of the unit ball in $\mathbb{R}^n$.
I've found Weyl's original work (Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen) but it is in German.
So is there an English translation or can anyone help? Thank you~
EDIT: Or, should this be a mathoverflow question?
Best Answer
Walter Strauss' book has a nice exposition of the proof. It uses comparison principles based on a variational characterization of the eigenvalues.