Weil asymptotic formula for the Laplacian on domains with rough boundaries

Abstract

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_α$-domain whenever $ (d-1)/α<d $, prove that the remainder in this formula is $ O(λ^{(d-1)/α}) $ and give an example where the remainder estimate $ O(λ^{(d-1)/α}) $ is order sharp. We use a new version of variational technique which does not require the extension theorem.