AI 中文总结
本文研究带小圆形孔的光滑有界区域中-Δ算子特征值与特征函数的定性性质，分析其特征函数定量估计、特征值单重性及节点集行为，关键用到u-电容势的点态估计。
AI 中文摘要
本文研究光滑有界区域Ω内带Dirichlet边界条件的-Δ算子的特征值与特征函数的定性性质，该区域含一个小圆形孔。文献中此情况称为“奇异摄动”，与“正则摄动”情况相对。记Ω_ε:=Ω\backslash B(P,ε)，其中B(P,ε)是中心在P、半径为ε的球，P∈Ω且ε足够小，我们研究：1) Ω_ε内-Δ算子特征函数的定量估计；2) Ω_ε内-Δ算子特征值的单重性；3) Ω_ε内-Δ算子特征函数的节点集行为。分析的关键要素是所谓u-电容势的点态估计，该势由文献[afhl]首次引入。
英文摘要
In this paper we study qualitative properties of the eigenvalues and eigenfunctions of $-Δ$ with Dirichlet boundary condition in a smooth bounded domain $Ω$ with a small circular hole. In the literature, this is known as a "singular perturbation", in contrast with the "regular perturbation" case. Denoting by $Ω_ε:=Ω\setminus B(P,ε)$ where $B(P,ε)$ is the ball centered at $P$ and radius $ε$, for $P\inΩ$ and $ε$ small enough we investigate 1) quantitative estimates for the eigenfunctions of $-Δ$ in $Ω_ε$; 2) the simplicity of the eigenvalues of $-Δ$ in $Ω_ε$; 3) the behavior of nodal sets of the eigenfunctions of $-Δ$ in $Ω_ε$. A key ingredient in our analysis consists of pointwise estimates on the so-called $u$-capacitary potential firstly introduced in \cite{afhl}.
