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We obtain estimates for the counting function of the Neumann Laplacian on a planar domain bounded by the graph of a lower semicontinuous L1-function. These estimates imply necessary and sufficient conditions for the validity of the classical one-term Weyl formula for the counting function and, under certain restrictions, give an order sharp remainder estimate in this formula.

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

We present explicit estimates for the remainder in the Weyl formula for the Laplace operator on a domain Ω, which involve only the most basic characteristics of Ω and hold under minimal assumptions about the boundary \(\partial \Omega\).