A decay estimate for the eigenvalues of the Neumann-Poincaré operator using the Grunsky coefficients

Jung, YoungHoon; Lim, Mikyoung

doi:10.1090/proc/14785

A decay estimate for the eigenvalues of the Neumann-Poincaré operator using the Grunsky coefficients
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Proc. Amer. Math. Soc. 148 (2020), 591-600 Request permission

Abstract:

We consider the decay property of the eigenvalues of the Neumann-Poincaré operator in two dimensions. As is well known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having $C^{1,\alpha }$ boundary with $\alpha \in (0,1)$. We show that the eigenvalues $\lambda _k$ of the Neumann-Poincaré operator ordered by size satisfy that $|\lambda _k| = O(k^{-p-\alpha +1/2})$ for an arbitrary simply connected domain having $C^{1+p,\alpha }$ boundary with $p\geq 0,~ \alpha \in (0,1)$, and $p+\alpha >\frac {1}{2}$.

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