The stiff Neumann problem: asymptotic specialty and "kissing" domains

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain $Ω\subset\mathbb{R}^d$ which is divided into two subdomains: an annulus $Ω_1$ and a core $Ω_0$. The density and the stiffness constants are of order $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ in $Ω_0$, while they are of order $1$ in $Ω_1$. Here $m\in\mathbb{R}$ is fixed and $\varepsilon>0$ is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as $\varepsilon \to 0$ for any $m$. In dimension $2$ the case when $Ω_0$ touches the exterior boudary $\partialΩ$ and $Ω_1$ gets two cusps at a point $\mathcal{O}$ 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 $x\to\mathcal{O}$ for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

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