Degenerate scales for thin elastic plates with Dirichlet boundary conditions

A degenerate scale occurs when a loss of uniqueness of the solution of the boundary integral equations happens for that scale of the problem. We consider here the biharmonic 2D problem with Dirichlet boundary conditions which models the bending behavior of a clamped isotropic Kirchhoff plate. We extend several results about degenerate scales previously found for the Laplace and Lame equation to the biharmonic equation in 2 dimensions. The degenerate scales are obtained from a 4x4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4$$\end{document} discriminant matrix whose shape is provided for different kinds of domain symmetry. We show that degenerate scales can be obtained from a minimization problem. Then, we compare the degenerate scales of two boundaries, one included within the other. For smooth star-shaped curves, we show that there are only two degenerate scales, give sufficient conditions for not being at a degenerate scale and produce bounds to the degenerate scales. For symmetric smooth simply connected curves there are also only two degenerate scales. For these symmetric cases, the use of complex variables allows us to go further and to link the problem of biharmonic equation to the one of plane elasticity and to give information, including bounds and exact values of degenerate scales, for many cases, while until now only very few ones were known. Our results have some consequences for the biharmonic outer radius defined by Polya and Szego. The numerical computation of degenerate scales by using boundary elements confirms the theoretical results.