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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Bibliographic Details
Published inGeometric and functional analysis Vol. 31; no. 5; pp. 1219 - 1244
Main Authors Logunov, A., Malinnikova, E., Nadirashvili, N., Nazarov, F.
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.10.2021
Springer Nature B.V
Subjects
Analysis
Dirichlet problem
Domains
Eigenvalues
Eigenvectors
Mathematics
Mathematics and Statistics
Smooth boundaries
Upper bounds
Laplace eigenfunction
Lipschitz domain
35B05
35J05
Nodal set
31B05
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-021-00581-5