Spectral structure of the Neumann-Poincar\'e operator on axially symmetric functions

TL;DR

This paper analyzes the spectral properties of the Neumann-Poincaré operator on axially symmetric 3D domains, revealing eigenfunction structures, asymptotics, and effects of boundary regularity and corners.

Contribution

It provides a detailed spectral analysis of the Neumann-Poincaré operator on axially symmetric domains, including eigenvalue asymptotics and the impact of boundary regularity.

Findings

Existence of infinitely many axially symmetric eigenfunctions.

Weyl-type asymptotics for eigenvalues on smooth domains.

Decay estimates of eigenvalues depending on boundary regularity.

Abstract

We consider the Neumann-Poincar\'e operator on a three-dimensional axially symmetric domain which is generated by rotating a planar domain around an axis which does not intersect the planar domain. We investigate its spectral structure when it is restricted to axially symmetric functions. If the boundary of the domain is smooth, we show that there are infinitely many axially symmetric eigenfunctions and derive Weyl-type asymptotics of the corresponding eigenvalues. We also derive the leading order terms of the asymptotic limits of positive and negative eigenvalues. The coefficients of the leading order terms are related to the convexity and concavity of the domain. If the boundary of the domain is less regular, we derive decay estimates of the eigenvalues. The decay rate depends on the regularity of the boundary. We also consider the domains with corners and prove that the essential spectrum of the Neumann-Poincar\'e operator on the axially symmetric three-dimensional domain is non-trivial and contains that of the planar domain.