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arXiv · 2312.14122

Dirichlet eigenfunctions with nonzero mean value

Stefan Steinerberger·Raghavendra Venkatraman
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Abstract

We consider Laplacian eigenfunctions on a domain $Ω\subset \mathbb{R}^d$. 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 $\mathbb{R}^d$, where among the first $n$ eigenfunctions only $\sim n^{1/d}$ have a mean value different from zero. We prove that this rate is sharp in \textit{any} smooth domain, up to a logarithmic factor: in any smooth domain~$Ω$, among the first $n$ Dirichlet eigenfunctions at least $ (\log{n})^{-1/2} \cdot n^{1/d} $ have a nonzero mean.

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Stefan Steinerberger, Raghavendra Venkatraman. 2025-03-15. Dirichlet eigenfunctions with nonzero mean value. https://arxiv.org/abs/2312.14122

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