Weil asymptotic formula for the Laplacian on domains with rough boundaries

Details Weil asymptotic formula for the Laplacian on domains with rough boundaries Weil asymptotic formula for the Laplacian on domains with rough boundaries

2003-10-07

arxiv.org/abs/math/0310093v1

Mathematics

Spectral Theory

32 pages

Scientific paper

We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in $ \R^n$. Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a $ \Lip_\alpha$-domain whenever $ (d-1)/\alpha

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