Dirichlet spectrum of the Fichera layer

Monique Dauge, Yvon Lafranche, Thomas Ourmières-Bonafos

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.

22 novembre 2017

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