Spectra of Two-Dimensional Models for Thin Plates with Sharp Edges

We investigate the spectrum of the two-dimensional model for a thin plate with a sharp edge. The model yields an elliptic $3\times3$ Agmon–Douglis–Nirenberg system on a planar domain with coefficients degenerating at the boundary. We prove that in the case of a degeneration rate $\alpha<2$, the spectrum is discrete, but, for $\alpha\geq2$, there appears a nontrivial essential spectrum. A first result for the degenerating scalar fourth order plate equation is due to Mikhlin. We also study the positive definiteness of the quadratic energy form and the necessity to impose stable boundary conditions. These results differ from the ones that Mikhlin published