A Strange Vertex Condition Coming From Nowhere

We prove norm-resolvent and spectral convergence in L-2 of solutions to the Neumann Poisson problem -Delta u(epsilon) = f on a domain Omega(epsilon) perforated by Dirichlet holes and shrinking to a 1 dimensional interval. The limit u satisfies an equation of the type -u '' + mu u = f on the interval (0, 1), where mu is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighborhood and the vertex neighborhood is chosen correctly, the constant it will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation.