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The essential spectrum of Neumann Laplacians on some bounded singular domains. (English) Zbl 0741.35043

The authors deal with the eigenvalue problem of Laplacian (with Neumann B.C.) in a singular domain and give a very interesting result on the structure of the spectrum and the geometry of the domain. If a bounded domain \(\Omega\) has a smooth boundary, \(-\Delta_ N^ \Omega\) has only point spectrums, which do not accumulate. However if \(\partial\Omega\) is singular, eigenvalues sometimes accumulate on some point, which becomes an essential spectrum. The domain the authors are dealing with has infinite slits (comb domain) and so the boundary \(\partial\Omega\) is very singular. Hence there arise essential spectrums. They construct a domain with such structure of spectrum. Moreover they give a method to control such essential spectrum by constructing the domain appropriately. They prove the following striking result. Theorem. For any closed set \(S\subset[0,\infty)\), there exists a bounded domain \(\Omega\) such that \(\sigma_{ess}(-\Delta_ N^ \Omega)=S\), \(\sigma_{ac}(-\Delta_ N^ \Omega)=\emptyset\).
Reviewer: S.Jimbo (Tsushima)

MSC:

35P05 General topics in linear spectral theory for PDEs
35R30 Inverse problems for PDEs
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