Spectral properties of the Neumann-Poincaré operator in 3D elasticity. (English) Zbl 1473.35100
Summary: We consider the adjoint double layer potential (Neumann-Poincaré (NP)) operator appearing in 3-dimensional elasticity. We show that the recent result about the polynomial compactness of this operator for the case of a homogeneous media follows without additional calculations from previous considerations by Agranovich et al., based upon pseudodifferential operators. Further on, we define the NP operator for the case of a nonhomogeneous isotropic media and show that its properties depend crucially on the character of nonhomogeneity. If the Lamé parameters are constant along the boundary, the NP operator is still polynomially compact. On the other hand, if these parameters are not constant, two or more intervals of continuous spectrum may appear, so the NP operator ceases to be polynomially compact. However, after a certain modification, it becomes polynomially compact again. Finally, we evaluate the rate of convergence of discrete eigenvalues of the NP operator to the tips of the essential spectrum.
MSC:
35C15 | Integral representations of solutions to PDEs |
35J57 | Boundary value problems for second-order elliptic systems |
35P05 | General topics in linear spectral theory for PDEs |
47B07 | Linear operators defined by compactness properties |
74B05 | Classical linear elasticity |