Spectral properties of the Neumann–Poincaré operator on rotationally symmetric domains

Abstract

This paper concerns the spectral properties of the Neumann–Poincaré operator on two- and three-dimensional bounded domains which are invariant under either rotation or reflection. We prove that if the domain has such symmetry, then the domain of definition of the Neumann–Poincaré operator is decomposed into invariant subspaces defined as eigenspaces of the unitary transformation corresponding to rotation or reflection. Thus, the spectrum of the Neumann–Poincaré operator is the union of those on invariant subspaces. In two dimensions, an m-fold rotationally symmetric simply connected domain D is realized as the mth-root transform of a domain, say $\Omega$. We prove that the spectrum on one of invariant subspaces is the exact copy of the spectrum on $\Omega$. It implies in particular that the spectrum on the transformed domain D contains the spectrum on the original domain $\Omega$ counting multiplicities. We present a matrix representation of the Neumann–Poincaré operator on the m-fold rotationally symmetric domain using the Grunsky coefficients. We also discuss some examples including lemniscates, m-star shaped domains and the Cassini oval.