The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

Let Ω be a bounded domain in R n with C 1 boundary and let u λ be a Dirichlet Laplace eigenfunction in Ω with eigenvalue λ . We show that the ( n - 1 ) -dimensional Hausdorff measure of the zero set of u λ does not exceed C ( Ω ) λ . This result is new even for the case of domains with C ∞ -smooth b...

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Veröffentlicht in:Geometric and functional analysis 2021-10, Vol.31 (5), p.1219-1244
Hauptverfasser: Logunov, A., Malinnikova, E., Nadirashvili, N., Nazarov, F.
Format: Artikel
Sprache:eng
Schlagworte:
Analysis
Dirichlet problem
Domains
Eigenvalues
Eigenvectors
Mathematics
Mathematics and Statistics
Smooth boundaries
Upper bounds
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Zusammenfassung:Let Ω be a bounded domain in R n with C 1 boundary and let u λ be a Dirichlet Laplace eigenfunction in Ω with eigenvalue λ . We show that the ( n - 1 ) -dimensional Hausdorff measure of the zero set of u λ does not exceed C ( Ω ) λ . This result is new even for the case of domains with C ∞ -smooth boundary.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-021-00581-5