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arXiv · 2009.10581

Density of zero sets for sums of eigenfunctions

Stefano Decio
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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.

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Stefano Decio. 2020-09-22. Density of zero sets for sums of eigenfunctions. https://doi.org/10.1080/03605302.2020.1857405

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