A decay estimate for the eigenvalues of the Neumann-Poincar\'{e} operator in two dimensions using the Grunsky coefficients

We investigate the decay property of the eigenvalues of the Neumann-Poincar\'{e} 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-Poincar\'{e} 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 $p+\alpha>\frac{1}{2}$.