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.

中文翻译：

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特征函数和的零集密度

我们考虑拉普拉斯-贝尔特拉米算子在紧凑黎曼流形 $(M,g)$ 上的本征函数的线性组合，并研究它们的零集的密度属​​性。更准确地说，令 $f=\sum_{k=1}^m a_k \phi_{\lambda_{j_k}}$，其中 $-\Delta_g\phi_{\lambda}=\lambda\phi_{\lambda}$。用 $Z_f$ 表示 $f$ 的零集，我们证明对于任何 $x\in M$，$dist(x,Z_f)\leq C(m)\lambda_{j_1}^{-1/2 }$。该证明基于高阶椭圆偏微分方程正解的新积分 Harnack 型估计。