Graph-like Domains with Dirichlet-Perforation

We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-Delta u_varepsilon = f$ on a domain $Omega_varepsilon$ perforated by Dirichlet-holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-uu0027u0027+mu u = f$ 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 is the scaling between the thickened edges and the thickened vertices is chosen correctly, the constant $mu$ will appear in the vertex condition of the limit problem.