A DECAY ESTIMATE FOR THE EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR USING THE GRUNSKY COEFFICIENTS

We consider the decay property of the eigenvalues of the NeumannPoincare 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,C-alpha boundary with alpha is an element of (0, 1). We show that the eigenvalues lambda(k) of the Neumann-Poincare operator ordered by size satisfy that vertical bar lambda(k)vertical bar = O(k(-p-alpha+1/2)) for an arbitrary simply connected domain having C-1+p,C-alpha boundary with p >= 0, alpha is an element of (0,1), and p + alpha > 1/2.