Year: 2025
Type: article
Abstract: We consider Laplacian eigenfunctions on a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d . Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary conditions: on ‘generic’ domains, one would expect that every eigenfunction has nonzero mean value. The other extreme is the ball in R d \mathbb {R}^d , where among the first n n eigenfunctions only ∼ n 1 / d \sim n^{1/d} have a mean value different from zero. We prove that this rate is sharp in any smooth domain, up to a logarithmic factor: in any smooth domain Ω \Omega , among the first n n Dirichlet eigenfunctions at least ( log n ) − 1 / 2 ⋅ n 1 / d (\log {n})^{-1/2} \cdot n^{1/d} have a nonzero mean.
DOI: 10.1090/proc/17372
Institutions: University of Washington, University of Utah
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4415443178