Aymptotics of eigenvalues of the Neumann-Poincar'e operator in 3D elasticity

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues of the Neumann-Poincaré operator in 3D elasticity, revealing their distribution near essential spectrum points and relating it to geometric properties of the surface.

Contribution

It provides the first detailed asymptotic analysis of eigenvalues for this operator in 3D elasticity, connecting spectral behavior to surface geometry.

Findings

Eigenvalues asymptotics near essential spectrum points

Eigenvalue distribution expressed via eigenvalue problem on cotangent bundle

Relation of eigenvalue distribution to Euler characteristics and Willmore energy

Abstract

We consider the Neumann-Poincar'e (double layer potential) operator in 3D elasicity on a smooth closed surface. Its essential spectrum consists of 3 points. We find the asymptotics of sequences of eigenvalues converging to these three pounts. They are expressed via the eigenvalue distribution for a matrix eigenvalue problem depending on the points of the cotangent bundle of the surface. For the two-sided asymptotics for the distribution of the the union of sequences of eigenvalues converging to the point of the essential spectrum from both sides, an expression is found via the Euler characteristics and the Willmore energy of the surface.