Density of zero sets for sums of eigenfunctions

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 $f={\sum }_{k=1}^m{a}_k{\varphi }_{{\lambda }_{j_k}}$, where $-{\Delta }_g{\varphi }_{\lambda }=\lambda {\varphi }_{\lambda }$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\le C(m){\lambda }_{j_1}^{-1/2}$. The proof is based on a new integral Harnack-type estimate for positive solutions of higher order elliptic PDEs.

21 pages