# How to prove Weyl’s asymptotic law for the eigenvalues of the Dirichlet Laplacian?

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?

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Thanks!${}{}{}$ –  t.b. Dec 6 '11 at 6:00