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

• Published in: Geometric and functional analysis Vol. 31; no. 5; pp. 1219 - 1244
• Main Authors: Logunov, A., Malinnikova, E., Nadirashvili, N., Nazarov, F.
• Format: Journal Article
• Language: English
• 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

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...