The Linearization of the Dirichlet-to-Neumann Map in the Anisotropic Kirchhoff–Love Plate Theory

Abstract

For each ${\bf C}$ elasticity tensor field over a two-dimensional bounded domain, we can naturally define the Dirichlet-to-Neumann map, denoted by $\Pi _{\bf C} $, which is an analogue of the voltage-to-current map in electrical impedance tomography (see for instance the survey paper [The Dirichlet-to-Neumann map and applications, Inverse Problems in PDE, Society for Industrial and Applied Mathematics, 1990] and the references therein). It is not known whether $\Pi _{\bf C} $ uniquely determines ${\bf C}$.In this paper, we are interested in the Frechet derivative of the map ${\bf C} \mapsto \Pi _{\bf C} $ at homogeneous ${\bf C}$, denoted by $d\Pi _{\bf C} $, and study the “size” of $\ker d\Pi _{\bf C} $. The main results are as follows: (1) If $\ker d\Pi _{\bf C} = 0$, the counting number of the set of all Stroh eigenvalues of ${\bf C}$ must be four. (2) If the counting number of the set of all Stroh eigenvalues of ${\bf C}$ is four and ${\bf C}$ satisfies an additional condition on its components ${\bf ...