Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

Rupert L. Frank; Simon Larson

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Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

Rupert L. Frank, Simon Larson

Spectral Theory · Submitted 28 Jan 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary.

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arXiv ID1901.09771 (version 2)

Pith IDJX535WPN, Pith's permanent record of this paper and its review

CitationRupert L. Frank, Simon Larson. "Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain." arXiv:1901.09771 (2019). https://pith.science/paper/JX535WPN

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.19 · mentions code or data availability 0.01.

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We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary.	result 0.55	0.96	0.88	asserted 1.00
s1	Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.	result 0.90	0.94	0.20	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original abstract

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

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1 arXiv paper cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set.

Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain · 0.83

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Compares with it 1 paper

We remark that Theorem [ref] reverses the Li-Yau inequalities and presents a positive correction term which, again, depends only on the measure of the boundary, hence complementing the results in [ref] (see also [ref] ).

Complementary asymptotically sharp estimates for eigenvalue means of Laplacians · Upper bounds with no restriction on the regularity of the boundary · 0.98

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