Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer

Abstract

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector $\kappa =({\kappa }_1,{\kappa }_2)$ which characterises the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases ${\kappa }_1\ge 0$ and ${\kappa }_2\in [-{\kappa }_1,{\kappa }_1]$. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $\kappa$. In particular, we show that for a given ${\kappa }_1>0$, there is some $h({\kappa }_1)>0$ such that discrete spectrum exists for ${\kappa }_2\in [-{\kappa }_1,0)\cup (h({\kappa }_1),{\kappa }_1]$ whereas it is empty for ${\kappa }_2\in [0,h({\kappa }_1)]$. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.