Spectral properties of the Neumann-Poincare operator on rotationally symmetric domains

초록

This paper concerns the spectral properties of the Neuman n-Poincare 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-Poincare operator is decomposed into invariant subspaces defined as eigenspaces of the unitary transformation corresponding to rotation or reflection. Thus, the spectrum of the Neumann-Poincard operator is the union of those on invariant subspaces. In two dimensions, an in-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-Poincard 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.

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