Planar Dirichlet eigenvalue bounds from a trace identity

Romain Speciel

Home/Archive/arXiv

arXiv · 2610.05636

Planar Dirichlet eigenvalue bounds from a trace identity

Romain Speciel

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove an asymptotically sharp lower bound for the eigenvalues $λ_k$ of the Dirichlet Laplacian on bounded open sets in the plane with area $A>0$. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain $(2/3)\cdot 4πk/A\leq λ_k$ for every $k$, which is two thirds of the bound conjectured by Pólya.

Explore related subjects

math.SP math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Romain Speciel. 2026-10-05. Planar Dirichlet eigenvalue bounds from a trace identity. https://arxiv.org/abs/2610.05636

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues

We prove the Ashbaugh--Benguria conjecture for bounded domains with Lipschitz boundary in $\mathbb R^m$. More precisely, among all bounded Lipschitz domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

math.SP↗

arXiv2026

Kuznecov formulas for domains with fractal boundary

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{λ_j\}_{j=0}^\infty$. For a domain $Ω\subset M$, consider $$ N_Ω(λ) = \sum_{λ_j\leqλ} \left|\int_Ωe_j\,dV_g\right|^2. $$ Suppose that $\partialΩ$ is $s$-Ahlfors regular, with $s\in{[d-1,d)}$, and that $Ω$ satisfies a two-sided corkscrew condition. We prove that $$ N_Ω(λ) = \operatorname{vol}(Ω)+O\bigl(λ^{s-d}\bigr), $$ and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order $λ^{s-d}$ holds precisely when the boundary admits a natural $s$-dimensional crossing volume $\mathcal V^s(\partialΩ)$, in which case $$ N_Ω(λ) = \operatorname{vol}(Ω) - C_{d,s}\, \mathcal V^s(\partialΩ)\, λ^{s-d} + o\bigl(λ^{s-d}\bigr). $$ For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.

math.SP↗

arXiv2026

On the Dirac Faber-Krahn Conjecture

In this paper, we prove the strict Faber-Krahn inequality for non-negative-mass Dirac operators on simply connected domains with infinite-mass boundary condition. Denoting the first positive eigenvalue of the corresponding Dirac operator with mass $m\geq 0$ as $λ_1^{+}(Ω;m)$, our main result shows that \begin{equation*} λ_1^{+}(Ω;m)\geq λ_1^{+}(B_{|Ω|};m), \end{equation*} where $B_{|Ω|}$ is the disk with the same area as the domain $Ω$, and the equality is strictly valid if and only if $Ω=B_{|Ω|}$. The key observation leading to the proof is that the upper component of the Dirac spinor associated with $λ_1^{+}(Ω;m)$ is non-vanishing over $Ω$, which is proved by exploiting its Dirac current and the associated stream function. Based on this global non-vanishing property, we construct a sphere-valued spinor map of degree one, which allows a sharp comparison with the radial disk profile using the planar isoperimetry.

math.SP↗