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 $\Omega\subset\mathbb{R}^d$ which is divided into two subdomains: an annulus $\Omega_1$ and a core $\Omega_0$. The density and the stiffness constants are of order $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ in $\Omega_0$, while they are of order $1$ in $\Omega_1$. 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 $\Omega_0$ touches the exterior boudary $\partial\Omega$ and $\Omega_1$ 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.