The stiff Neumann problem: Asymptotic specialty and “kissing” domains
Valeria Chiadò Piat, Lorenza D’Elia, С. А. Назаров · Asymptotic Analysis · 2021
We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain [Formula: see text] which is divided into two subdomains: an annulus [Formula: see text] and a core [Formula: see text]. The density and the stiffness constants are of order [Formula: see text] and [Formula: see text] in [Formula: see text], while they are of order 1 in [Formula: see text]. Here [Formula: see text] is fixed and [Formula: see text] is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as [Formula: see text] for any m. In dimension 2 the case when [Formula: see text] touches the exterior boundary [Formula: see text] and [Formula: see text] gets two cusps at a point [Formula: see text] 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 [Formula: see text] for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.