Dirichlet spectrum of the Fichera layer

Monique Dauge (1) , Yvon Lafranche (1) , Thomas Ourmières-Bonafos (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMO - Laboratoire de Mathématiques d'Orsay

We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Such a domain contains six edges and two corners. It is a canonical example of what is called a non-smooth conical layer and we name it after Fichera because near the non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. We show that the essential spectrum of the Laplacian on such a domain is a half-line and we characterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm-Liouville operator. By finite element computations, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Analyse numérique [math.NA]

Monique Dauge, Yvon Lafranche, Thomas Ourmières-Bonafos. Dirichlet spectrum of the Fichera layer. 2017. ⟨hal-01645083v1⟩