Weyl 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.

Gaussian free fields for mathematicians

“…(References and much more precise estimates on the growth of λ j are given in [NS05].) We define powers of the negative Dirichlet Laplacian by writing, for each a ∈ R,…”

Two scale Hardy space infinite elements for scalar waveguide problems

“…Using Weyl asymptotics λ n = O(n 2 d−1 ) (see e.g. [25] and the references therein) on the interface ϒ ⊂ R d−1 and by definition of κ n with θ ∈ (π, 2π), (iκ n (ω)) = O(− √ λ n ) holds for sufficiently large n ∈ N. Hence, for ξ > 0 the sequence u n (ξ ) = c n exp(iκ n (ω)ξ ) with Fourier coefficients c n := u(0, •), ϕ n L 2 (ϒ) decays exponentially (d = 2) or at least super algebraically (d = 3) for n → ∞ . If a part of the waveguide with length L is added to the interior domain, the new constantsc n := u(L, •), ϕ n L 2 (ϒ) are given by c n exp(iκ n (ω)L) and decay sufficiently fast.…”

Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian

“…In general these estimates will depend on Ω, and C count will not be independent of Ω (see e.g. [26,34]). …”

“…This in turn only occurs if there are boundary points where the Wiener series (for the boundary) converges [55,28]. For the Neumann case the situation is more complicated [34,26,33]. In particular, there are domains with arbitrary closed continuous Neumann spectrum [26].…”