Singularity of eigenfunctions at the junction of shrinking tubes. I (Q2434787)

The paper deals with the behavior of eigenfunctions of the Dirichlet Laplacian on a dumbbell domain formed by two slightly different domains connected by a thin tube, for which the mass of a given eigenfunction concentrates in one of them. The restriction of the eigenfunction to the other domain develops a singularity at the junction of the tube, when the channel section tends to zero. Based upon a sharp control of the transversal frequencies along the connecting tube, the authors evaluate the rate of the singularity at this junction and they prove that the solution is one-sign in a neighborhood of the singular junction.