计算与解析域边界接触的节点线
Counting Nodal Lines Which Touch the Boundary of an Analytic Domain
摘要：
本文考虑实解析平面区域$Ω$的Neumann特征函数$ϕ_λ$在边界$\partial Ω$上的零点.我们证明了它的边界零点的个数是$O (λ)$，其中$-Δϕ_λ = λ^2 ϕ_λ$ .我们还证明了Neumann或Dirichlet本征函数的边界临界点的个数是$O(λ)$。因此，接触边界的$ϕ_λ$（节点集的分量）的节点线的数量是$λ$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。
英文摘要：
We consider the zeros on the boundary $\partial Ω$ of a Neumann eigenfunction $ϕ_λ$ of a real analytic plane domain $Ω$. We prove that the number of its boundary zeros is $O (λ)$ where $-Δϕ_λ = λ^2 ϕ_λ$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(λ)$. It follows that the number of nodal lines of $ϕ_λ$ (components of the nodal set) which touch the boundary is of order $λ$. 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.
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