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 of 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 epsilon(-2m) and epsilon(-1) in Omega(0), while they are of order 1 in( )Omega(1). Here m is an element of R is fixed and epsilon > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as epsilon -> 0 for any m. In dimension 2 the case when Omega(0) touches the exterior boundary partial derivative Omega S and Omega(1) 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.