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

Abstract

We investigate 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 $\alpha \in (0,1)$. In this paper, we show that the eigenvalue ${\lambda }_k$'s 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 $p\ge 0,\alpha \in (0,1)$ and $p+\alpha >\frac{1}{2}$.

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• Convergence rate for eigenvalues of the elastic Neumann--Poincaré operator on smooth and real analytic boundaries in two dimensions