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How do I show that the asymptotic speed of the eigenvalues $\lambda$ of the Laplacian Operator is $O(m^{2/n})$ where $m$ is the index of the eigenvalues and $n$ is the dimension of the space?

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This is not a perfect answer, but in the book by J. Roe, "Elliptic operators, topology, and asymptotic methods" there is at the start of chapter 8 a rough estimate along the direction you want. Maybe by using fractional Sobolev exponent and some more thinking it can give what you are after. In the same book later there is a full proof of Weyl's asymptotic formula (which is a stronger statement than what you are after).

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  • $\begingroup$ Does this proof require the smoothness of the boundary? What if the boundary is nowhere differentiable like a fractal curve? $\endgroup$ – Hans Jan 6 at 21:05

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