A Strange Vertex Condition Coming from Nowhere

Abstract

We prove norm-resolvent and spectral convergence in L 2 of solutions to the Neumann Poisson problem - \Delta u\varepsi = f on a domain \Omega \varepsi perforated by Dirichlet holes and shrinking to a 1dimensional interval. The limit u satisfies an equation of the type - u \prime \prime + \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 \mu 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.