Dirichlet Spectrum of the Fichera Layer

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.2010 Mathematics Subject Classification. 35J05, 35P15, 35Q40, 81Q10, 65M60.

Dirac operator spectrum in tubes and layers with a zigzag type boundary

“…In contrast to the two-dimensional situation little is known about spectral properties of sharply broken layers. Using the result of [DLO18] we can make a claim about the 'octant', or 'Fichera' layer, which can be regarded as a counterpart to the L-shaped strip, namely the region Ω := {(x, y, z) ∈ R 2 : x, y, z > 0, min(x, y, z) < π}.…”

“…In[DLO18] a numerical argument is used to show that the discrete spectrum is nonempty. 3 This result in[EK15] contains the same misprint, an extra square of f .…”

Spectral Asymptotics of the Dirichlet Laplacian on a Generalized Parabolic Layer

“…In this perspective, more precise analysis of various particular cases deserves attention. Examples of layers treated so far include mildly curved layers [BEGK01,EKr01], conical layers [DOR15,ET10,OP17], and so-called octant (or Fichera) layers [DLO17].…”

Spectral analysis in broken sheared waveguides

“…Spectral properties of the Dirichlet Laplacian operator in unbounded domains of the n dimensional Euclidean space has been extensively studied in the last years. In this context, the question on the existence of discrete spectrum naturally arises; the results are non-trivial and depend on the geometry of the region [1,2,3,4,5,6,7,8,10,11,12,13,14,15,18,19,20,21,22,24,25,26,27,28,30]. For example, consider a tubular neighborhood of a reference curve in R 3 .…”