The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

Abstract

For the Neumann-Poincaré (double layer potential) operator in the three-dimensional elasticity we establish asymptotic formulas for eigenvalues converging to the points of the essential spectrum and discuss geometric and mechanical meaning of coefficients in these formulas. In particular, we establish that for any body, there are infinitely many eigenvalues converging from above to each point of the essential spectrum. On the other hand, if there is a point where the boundary is concave (in particular, if the body contains cavities) then for each point of the essential spectrum there exists a sequence of eigenvalues converging to this point from below.