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Dan Mangoubi

Publications and source records attributed to Dan Mangoubi.

At least 19 recordsLinked to original sources
arXiv2026

A Vibrating Clamped Plate Resembles a Vibrating Membrane at High Frequency

We study the clamped plate eigenvalue problem. We show that clamped plate eigenfunctions can be approximated in the $L^2$-sense by their oscillatory components, which solve the Helmholtz equation, with an error that decays exponentially fast in the frequency on any compact subdomain. Additionally, we establish an upper bound on the boundary localization of clamped plate eigenfunctions analogous to the one observed for membrane eigenfunctions.

math.AP↗
arXiv2025

On multiplicity bounds for eigenvalues of the clamped round plate

We ask whether the only multiplicities in the spectrum of the clamped round plate are trivial, i.e., whether all existing multiplicities are due to the isometries of the sphere, or, equivalently, whether any eigenfunction is separated. We prove that any eigenfunction can be expressed as a sum of at most two separated ones, by showing that otherwise the corresponding eigenvalue is algebraic, contradicting the Siegel-Shidlovskii theory. In two dimensions it follows that no eigenvalue is of multiplicity greater than four. The proof exploits a linear recursion of order two for cross-product Bessel functions with coefficients which are not even algebraic functions, though they do satisfy a non-linear algebraic recursion.

math.SP↗
arXiv2024

The inner radius of nodal domains in high dimensions

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $λ$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $cλ^{-1/2}(\logλ)^{-(d-2)/2}$. This ball is centered at a point at which the eigenfunction attains its maximum in absolute value within the nodal domain.

math.AP↗
arXiv2021

Strong convexity for harmonic functions on compact symmetric spaces

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $Δ^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $Δ$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on $\mathbb{R}^n$ discovered by the first two authors and is related to strong convexity of the $L^2$-growth function of harmonic functions.

math.SP↗
arXiv2018

Harmonic functions vanishing on a cone

Let $Z$ be a quadratic harmonic cone in $\mathbb{R}^{3}$. We consider the family $\mathcal{H}(Z)$ of all harmonic functions vanishing on $Z$. Is $\mathcal{H}(Z)$ finite or infinite dimensional? Some aspects of this question go back to as early as the 19th century. To the best of our knowledge, no nondegenerate quadratic harmonic cone exists for which the answer to this question is known. In this paper we study the right circular harmonic cone and give evidence that the family of harmonic functions vanishing on it is, maybe surprisingly, finite dimensional. We introduce an arithmetic method to handle this question which extends ideas of Holt and Ille and is reminiscent of Hensel's Lemma.

math.AP↗
arXiv2015

On the sharpness of a three circles theorem for discrete harmonic functions

Any three circles theorem for discrete harmonic functions must contain an inherent error term. In this paper we find the sharp error term in an $L^2$-three circles theorem for harmonic functions defined in $\Zb^2$. The proof is highly indirect due to combinatorial obstacles and cancellations phenomena. We exploit Newton interpolation methods and recursive arguments.

math.CA↗
arXiv2013

Li-Yau inequality on graphs

We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute this curvature for lattices and trees and conclude that it behaves more naturally than the already existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth. We also derive Harnack inequalities and heat kernel bounds from the gradient estimate, and show how it can be used to strengthen the classical Buser inequality relating the spectral gap and the Cheeger constant of a graph.

math.AP↗
arXiv2013

Nodal geometry of graphs on surfaces

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the $n$th Laplacian eigenvalue is at most $2\left[ 6(n-1) + 15(2g-2) \right]^2$. Our results hold for any Schrödinger operator, not just the Laplacian.

math.CO↗
arXiv2013

A gradient estimate for harmonic functions sharing the same zeros

Let u, v be two harmonic functions in the disk of radius two which have exactly the same set Z of zeros. We observe that the gradient of \log |u/v| is bounded in the unit disk by a constant which depends on Z only. In case Z is empty this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives Hölder estimates on \log |u/v|.

math.AP↗
arXiv2012

The effect of curvature on convexity properties of harmonic functions and eigenfunctions

We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.

math.AP↗
arXiv2010

A Remark on Recent Lower Bounds for Nodal Sets

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for eigenfunctions and a growth vs. volume relation we proved in a previous work.

math.AP↗
arXiv2010

The Volume of a Local Nodal Domain

Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.

math.SP↗
arXiv2008

Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.

math.SP↗
arXiv2008

Local Asymmetry and the Inner Radius of Nodal Domains

Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/λ^n. We apply this result to prove that each nodal domain contains a ball of radius > C/λ^n.

math.SP↗
arXiv2007

Spectral Flexibility of Symplectic Manifolds T^2 x M

We consider Riemannian metrics compatible with the natural symplectic structure on T^2 x M, where T^2 is a symplectic 2-Torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ_1. We show that λ_1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension >= 4. We reduce the general conjecture to a purely symplectic question.

math.SP↗