刚性圆盘二维正方形堆积谱中的间隙
Gaps in the spectrum of two-dimensional square packing of stiff disks
摘要：
在本文中，我们调查通过渐近方法的开放间隙的频谱的刚性问题的拉普拉斯算子$-Δ$在$\mathbb{R}^2$穿孔连续圆孔。在含有$m\in (0,1/2)$的孔中，密度和刚度常数为$\varepsilon^{-2m}$和$\varepsilon^{-1}$阶。给出了与第一类贝塞尔函数有关的本征值和相应本征函数的首项的显式表达式。
英文摘要：
In this paper we investigate via an asymptotic method the opening of gaps in the spectrum of a stiff problem for the Laplace operator $-Δ$ in $\mathbb{R}^2$ perforated by contiguous circular holes. The density and the stiffness constants are of order $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ in the holes with $m\in (0,1/2)$. We provide an explicit expression of the leading terms of the eigenvalues and the corresponding eigenfunctions which are related to the Bessel functions of the first kind.
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