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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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2 Answers

up vote 2 down vote accepted

I've found a related question here, and it was answered. Its answer gives a book where an English proof can be found: Elliptic operators, topology, and asymptotic method by J. Roe.

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Walter Strauss' book has a nice exposition of the proof. It uses comparison principles based on a variational characterization of the eigenvalues.

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Walter Strauss's book? This one? –  t.b. Dec 6 '11 at 5:55
    
Close but not quite. I included a link. –  timur Dec 6 '11 at 5:59
    
Thanks!${}{}{}$ –  t.b. Dec 6 '11 at 6:00
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