The convergence of the spectrum of a weakly connected domain

The paper deals with the convergence, as ε tends to zero, of the spectrum of the Neumann problem -Δυɛ=λ(ɛ)υɛ in a «weakly connected» periodic domain Ωɛ of ℝ3. The domain Ωɛ is composed of a finite number of disjoint connected domains linked by thin bridges (curved plates or tubes). Under a few assumptions on the characteristic sizes of these bridges, we give an explicit asymptotic formula for the eigenvalues which tend to zero and we prove that the rest of the spectrum converges to the spectrum of an elliptic coupled system.