The first Hadamard variation of Neumann-Poincar\'e eigenvalues on the sphere

TL;DR

This paper investigates the behavior of Neumann-Poincaré eigenvalues on the sphere under domain deformation, revealing a zero derivative in their bifurcations and exploring related conjectures.

Contribution

It introduces the first Hadamard variation of Neumann-Poincaré eigenvalues on the sphere and analyzes their bifurcation properties under domain deformation.

Findings

The eigenvalues are explicitly given as 1/(2(2k+1)) with multiplicity 2k+1.

The Fréchet derivative of the sum of bifurcations is zero.

Connections to conjectures about the Neumann-Poincaré operator are discussed.

Abstract

The Neumann-Poincar\'e operator on the sphere has $\frac{1}{2(2k+1)}$, $k=0,1,2,\ldots$, as its eigenvalues and the corresponding multiplicity is $2k+1$. We consider the bifurcation of eigenvalues under deformation of domains, and show that Frech\'et derivative of the sum of the bifurcations is zero. We then discuss the connection of this result with some conjectures regarding the Neumann-Poincar\'e operator.