Counting Nodal Lines Which Touch the Boundary of an Analytic Domain
Counting Nodal Lines Which Touch the Boundary of an Analytic Domain
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计算接触分析域边界的节点线
DOI：
10.4310/jdg/1236604347
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
S. Zelditch
中科院分区：
文献类型：
--
作者：
J. Toth;S. Zelditch
我们考虑一个真实的解析平面域$\Omega$的Neumann特征函数$\phi_{\lambda}$的边界$\partial \Omega $上的零点。我们证明了它的边界零点的个数是O（\lambda）$其中$-\Delta \phi_{\lambda} =\Delta ^2\phi_{\lambda}$。我们还证明了Neumann或Dirichlet特征函数的边界临界点的数目是O（\lambda）。由此可见，$\phi_{\lambda}$（节点集的分量）中与边界接触的节点线的数量为$\lambda$阶。这个上界与总节线的长度具有相同的数量级，但是是内部节点分量数量的柯朗界的平方根。更一般地，证明了分段解析域的结果。
We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(\lambda)$. It follows that the number of nodal lines of $\phi_{\lambda}$ (components of the nodal set) which touch the boundary is of order $\lambda$. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.