We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Ω⊂Rd which is divided into two subdomains: an annulus Ω1 and a core Ω0. The density and the stiffness constants are of order ε−2m and ε−1 in Ω0, while they are of order 1 in Ω1. Here m∈R is fixed and ε>0is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as ε→0 for any m. In dimension 2 the case when Ω0 touches the exterior boundary ∂Ω and Ω1 gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x→O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

中文翻译：

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僵化的诺伊曼问题：渐近专业和“接吻”域

我们在光滑有界域ΩRd中研究了Laplace算子的刚性谱Neumann问题，该域分为两个子域：环Ω1和核心Ω0。密度和刚度常数在Ω0中约为ε-2m和ε-1，而在Ω1中则约为1。这里m∈R是固定的，并且ε> 0小。对于任何m，我们为特征值和相应的特征函数提供渐近性，如ε→0。在维度2中，考虑了Ω0接触外部边界ΩΩ且Ω1在点O处出现两个尖点的情况。采用与“平滑”情况相同的渐近程序的可能性是基于不规则部分附近的本征函数的结构。给出了在尖峰域中Laplace算子的混合边值问题的解的从x到O的完整渐近级数。