论文标题
Courant的Nodal域定理的本地版本
Local version of Courant's nodal domain theorem
论文作者
论文摘要
令$（m，g）$为封闭的riemannian歧管，其中g为$ c^1 $ - 平滑度量。考虑M $上的Laplace操作员的本征$ u_k $ u_k $的顺序。我们证明了与$ b $相交的$ u_k $的节点域数量的详尽估计。唐纳利（Donnelly）和费弗曼（Fefferman）提出了一个想法，即人们如何证明这种界限，他们也提出了局部界限和节点域数量的局部界限问题。我们将他们的思想与两种成分结合在一起：最近的征征型不平等和狭窄域中的Landis型生长引理。
Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of $u_k$ that intersect $B$. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.
