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

TL;DR

This paper establishes sufficient conditions for the discrete spectrum of the biharmonic operator in a peak-shaped domain, considering various boundary conditions and the domain's sharpness, advancing spectral analysis of Kirchhoff plates.

Contribution

It introduces new criteria linking boundary conditions and peak sharpness to the spectral properties of Kirchhoff plates with infinite peaks.

Findings

Discrete spectrum depends on boundary conditions and peak sharpness

Conditions vary with different boundary conditions

Results provide a spectral characterization for peak-shaped domains

Abstract

Sufficient conditions for a discrete spectrum of the biharmonic equation in a two-dimensional peak-shaped domain are established. Different boundary conditions from Kirchhoff's plate theory are imposed on the boundary and the results depend both on the type of boundary conditions and the sharpness exponent of the peak.