Weil asymptotic formula for the Laplacian on domains with rough boundaries

2003/10/07 by Yu. Netrusov, Yu. V. Netrusov, Netrusov, Yu. +2 · 1 citation
Computer Science · Mathematics · #35J20 #35P20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35J20 #msc:35P20

paper · pdf · doi:10.48550/arxiv.math/0310093

32 pages

arxiv created 2003/10/07 · openalex publication_date 2003/10/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \Lip_α-domain whenever (d-1)/α

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• Pólya's conjecture up to ε-loss and quantitative estimates for the remainder of Weyl's law

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