Dirichlet spectrum of the Fichera layer

TL;DR

This paper analyzes the spectral properties of the Dirichlet Laplacian on a three-dimensional Fichera layer, revealing the essential spectrum, finite discrete spectrum, and specific eigenvalues through theoretical and numerical methods.

Contribution

It provides a detailed spectral analysis of the Fichera layer, including the essential spectrum characterization, finiteness of discrete eigenvalues, and numerical identification of eigenvalues, extending to variants with rounded edges.

Findings

Essential spectrum is a half-line with a known minimum.

Finite number of discrete eigenvalues below the essential spectrum.

Exactly one eigenvalue found below the essential spectrum threshold.

Abstract

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. Alternatively, this domain can be viewed as an octant from which another "parallel" octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer. 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 taking singularities into account, 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.