A sufficient condition for a discrete spectrum of the Kirchhoff plate with an infinite peak

Abstract

Elliptic boundary value problems on domains which have a Lipschitz boundary and a compact closure, in particular when they generate positive self-adjoint operators, have fully discrete spectra. However, if the domain loses the Lipschitz property or compactness, other situations may occur. It is well-known that for the Dirichlet case boundedness is sufficient but not necessary for a discrete spectrum. See the famous paper by Rellich [26] or the more recent contributions by [27, 3]. On the other hand, for the Neumann problem of the Laplace operator there exist numerous examples of bounded domains such that the spectrum gets a non-empty continuous component (see e.g. [7, 17, 18, 28, 11]). The literature on the spectra for the Laplace operator with various boundary conditions is focussed on domains that have a cusp, a finite or infinite peak or horn [3, 11, 13, 8, 14, 12, 6, 4, 15] or even considered a rolled horn [28]. The criteria in [1] and [9] for the embedding H1(Ω) ⊂ L2(Ω) to be compact, show that the Neumann-Laplace problem on a domain Ω with the infinite peak