Density of zero sets for sums of eigenfunctions

TL;DR

This paper studies the distribution of zeros of linear combinations of Laplace eigenfunctions on compact manifolds, establishing a bound on how close any point can be to the zero set based on eigenvalues.

Contribution

It introduces a new integral Harnack-type estimate for higher order elliptic PDEs and applies it to analyze zero set density of eigenfunction combinations.

Findings

Zero sets are within a distance proportional to the inverse square root of the eigenvalue.

Established a uniform bound on the proximity of points to the zero set.

Provided a novel PDE estimate applicable to eigenfunction analysis.

Abstract

We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$ and investigate a density property of their zero sets. More precisely, let $f=\sum_{k=1}^m a_k \phi_{\lambda_{j_k}}$, where $-\Delta_g\phi_{\lambda}=\lambda\phi_{\lambda}$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\leq C(m)\lambda_{j_1}^{-1/2}$. The proof is based on a new integral Harnack-type estimate for positive solutions of higher order elliptic PDEs.