New asymptotic effects for the spectrum of problems on concentrated masses near the boundary
С. А. Назаров, Eugenia Pérez · Comptes Rendus Mécanique · 2009
The Dirichlet and Neumann spectral problems for the Laplace operator in a bounded domain Ω ⊂ R 2 are considered. We assume that Ω has a piecewise smooth boundary ∂ Ω and the density function is equal to 1 + ε − m χ ε in Ω , where ε > 0 is a small parameter, m ∈ R and χ ε is the characteristic function of the union ω ε 0 ∪ ⋯ ∪ ω ε J − 1 of small sets (the concentrated masses) distributed periodically near a straight segment Γ ⊂ ∂ Ω . We describe asymptotics for the eigenelements of both problems as ε → 0 .