Steklov问题的“闪烁特征值”在尖点区域产生连续谱
"Blinking eigenvalues" of the Steklov problem generate the continuous spectrum in a cuspidal domain
摘要：
研究Laplace算子在有界域$Ω\subset \mathbb{R}^d$，$d \geq 2$中的Steklov谱问题，其中有一个尖点使得该问题的连续谱非空，以及在一族有界域$Ω^\varepsilon \subset Ω$，$\varepsilon > 0$中的Steklov谱问题，其中有界域$Ω^\varepsilon \subset Ω$，$\varepsilon > 0$是由$Ω$通过在$\varepsilon$到尖点的距离处钝化尖点而得到的.虽然钝化域$Ω^\varepsilon$中的谱由本征值的无界正序列$\{ λ_j^\varepsilon \}_{j=1}^\infty$的固定$\varepsilon$组成，但我们将某些本征值的不同类型的行为挑选为$\varepsilon \to +0$：特别地，我们发现了稳定的、闪烁的和滑动的特征值族，我们还描述了一种将特征值序列族转化为$Ω$中问题的连续谱的机制，当$\varepsilon \to +0$。
英文摘要：
We study the Steklov spectral problem for the Laplace operator in a bounded domain $Ω\subset \mathbb{R}^d$, $d \geq 2$, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains $Ω^\varepsilon \subset Ω$, $\varepsilon > 0$, obtained from $Ω$ by blunting the cusp at the distance of $\varepsilon$ from the cusp tip. While the spectrum in the blunted domain $Ω^\varepsilon$ consists for a fixed $\varepsilon$ of an unbounded positive sequence $\{ λ_j^\varepsilon \}_{j=1}^\infty$ of eigenvalues, we single out different types of behavior of some eigenvalues as $\varepsilon \to +0$: in particular, stable, blinking, and gliding families of eigenvalues are found. We also describe a mechanism which transforms the family of the eigenvalue sequences into the continuous spectrum of the problem in $Ω$, when $\varepsilon \to +0$.
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