The Discrete Spectrum of the Neumann-poincare Operator in 3D Elasticity.

Abstract For the Neumann-Poincare (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 coeffcients 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. The reasoning is based upon the representation of the Neumann-Poincare operator as a zero order pseudo-differential operator on the boundary and on the earlier results by the author on the eigenvalue asymptotics for polynomially compact pseudodifferential operators. 2010 Mathematics Subject Classification. 47A75 (primary), 58J50 (secondary).