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

Kazunori Ando; Hyeonbae Kang; Yoshihisa Miyanishi; Erika Ushikoshi

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arXiv · 1805.02414

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

Kazunori Ando·Hyeonbae Kang·Yoshihisa Miyanishi·Erika Ushikoshi

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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.

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Kazunori Ando, Hyeonbae Kang, Yoshihisa Miyanishi, Erika Ushikoshi. 2018-05-07. The first Hadamard variation of Neumann-Poincar\'e eigenvalues on the sphere. https://arxiv.org/abs/1805.02414

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