Weyl Asymptotic Formula for the Laplacian on Domains with Rough Boundaries

By Y. Netrusov, Y. Safarov. 2003

82 citations recorded by the source corpus.

We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in ℝn. 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 Lipα-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 the variational technique which does not require the extension theorem.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/2ba98931d83b7449ba1eb53e607ec85e668c8065

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Universal Local Parametrizations via Heat Kernels and Eigenfunctions of the Laplacian (2007, background): “For the Neumann case the situation is more complicated [36, 27, 35].”
Gaussian free fields for mathematicians (2003, background): “en d= 2.) The reader may recall that by Weyl’s formula, if D⊂ Rd is bounded, then j2/d/(−λ j) tends to a constant as j→ ∞. (References and much more precise estimates on the growth of λj are given in [NS05].) We deﬁne powers of the negative Dirichlet Laplacian by writing, for each a∈ R, (−∆)a X βjej”
Geometrical Structure of Laplacian Eigenfunctions (2012, methodology): “Safarov, Weyl Asymptotic Formula for the Laplacian on Domains with Rough Boundaries, Commun.”
Maximization of the second positive Neumann eigenvalue for planar domains (2008, background): “This is true, for instance, if satisfies the cone condition, that is there are no outward pointing cusps (see [ NS ] for more refined conditions and a detailed discussion).”