论文标题
特征函数和零集的密度
Density of zero sets for sums of eigenfunctions
论文作者
论文摘要
我们考虑了紧凑的Riemannian歧管$（M，G）$的Laplace-Beltrami操作员特征函数的线性组合，并研究其零集的密度属性。更准确地说，令$ f = \ sum_ {k = 1}^m a_k ϕ_ {λ_{j_k}} $，其中$-Δ_GDICTION_λ= λϕ_λ $。用$ z_f $表示$ f $的零集，我们表明，对于任何$ x \ in m $，$ dist（x，z_f）\ leq c（m）λ__{J_1}^{ - 1/2} $。该证明是基于对高阶椭圆pdes积极解决方案的新的积分harnack型估计。
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 $f=\sum_{k=1}^m a_k ϕ_{λ_{j_k}}$, where $-Δ_gϕ_λ=λϕ_λ$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\leq C(m)λ_{j_1}^{-1/2}$. The proof is based on a new integral Harnack-type estimate for positive solutions of higher order elliptic PDEs.
