Dirichlet eigenfunctions with nonzero mean value

TL;DR

This work analyzes Dirichlet Laplacian eigenfunctions on a smooth bounded domain $Ω$ and quantifies how many of the first $n$ eigenfunctions have nonzero mean $∫_Ω φ_k$. Employing a heat-flow strategy, the authors construct a boundary-layer test function and relate boundary-mass loss to the presence of nonzero-mean eigenfunctions, leveraging Hassell–Tao bounds and Weyl's law. They prove a lower bound of order $n^{1/d}/√{log n}$ for the count of nonzero-mean eigenfunctions among the first $n$, and show that this rate is sharp up to the logarithmic factor on general smooth domains. The results illuminate the distribution of mean values in Dirichlet spectra and have connections to photonics and divergence-form PDEs with high-contrast coefficients, where these eigenfunctions play a role in resonator and waveguide designs.

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.

Theorems & Definitions (15)

• Proposition
• proof
• Theorem
• Lemma 1
• proof : Proof of Lemma 1
• Lemma 2: Reflection principle; e.g. katz
• Lemma 3
• proof
• Lemma 4: Hassell-Tao hassell
• proof