Spectral structure of the Neumann-Poincar\'e operator on thin ellipsoids and flat domains

TL;DR

This paper analyzes how the eigenvalues of the Neumann-Poincaré operator distribute on thin ellipsoids, revealing dense spectra in specific intervals as these ellipsoids become more elongated or flatter, with implications for flat 3D domains.

Contribution

It demonstrates the dense distribution of eigenvalues of the Neumann-Poincaré operator on thin ellipsoids and flat domains, highlighting how spectral properties evolve with shape deformation.

Findings

Eigenvalues densely distributed in [0,1/2] for long prolate ellipsoids

Eigenvalues densely distributed in [-1/2, 1/2] for flat oblate ellipsoids

More negative eigenvalues appear as oblate ellipsoids become flatter

Abstract

We investigate the spectral structure of the Neumann-Poincar\'e operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate ellipsoids and flat oblate ellipsoids. We show that the totality of eigenvalues of the Neumann-Poincar\'e operators on a sequence of the prolate spheroids is densely distributed in the interval [0,1/2] as their eccentricities tend to 1, namely, as they become longer. We then prove that eigenvalues of the Neumann-Poincar\'e operators on the oblate ellipsoids are densely distributed in the interval [-1/2, 1/2] as the ellipsoids become flatter. In particular, this shows that even if there are at most finitely many negative eigenvalues on the oblate ellipsoids, more and more negative eigenvalues appear as the ellipsoids become flatter. We also show a similar spectral property for flat three dimensional domains.