A decay estimate for the eigenvalues of the Neumann-Poincaré operator using the Grunsky coefficients
Younghoon Jung, Mikyoung Lim · Proceedings of the American Mathematical Society · 2019
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 , α C^{1,\alpha } boundary with α ∈ ( 0 , 1 ) \alpha \in (0,1) . We show that the eigenvalues λ k \lambda _k of the Neumann-Poincaré operator ordered by size satisfy that | λ k | = O ( k − p − α + 1 / 2 ) |\lambda _k| = O(k^{-p-\alpha +1/2}) for an arbitrary simply connected domain having C 1 + p , α C^{1+p,\alpha } boundary with p ≥ 0 , α ∈ ( 0 , 1 ) p\geq 0,~ \alpha \in (0,1) , and p + α > 1 2 p+\alpha >\frac {1}{2} .