Eigenvalues of the double layer potential

Question

Consider the double layer potential $K: L^2(S^2)\to L^2(S^2)$ $$(Kf)(x)=\int_{S^2}f(y)\frac{\partial}{\partial v_y}E(x,y)dS_y,$$ where $E(x,y)=||x-y||^{-1}$ and $\frac{\partial}{\partial v_y}$ means the outer normal derivative on $S^2$.

Show that the eigenvalues of $K$ are $\lambda_k=\frac{-1}{2k+1}$, $k=0,1,...$.

Remark: (1) In the paper (http://www.sciencedirect.com/science/article/pii/0022247X86902556), the authors said on p.2 that ''by a straightforward calculation it can be shown that the eigenvalues are given by $\lambda_k=\frac{-1}{2k+1}$''.

(2)In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X99965381), on p.2 we can see that the eigenfunctions of $K$ are spherical harmonics.

Answer 1

This is a classic problem solved by Poincaré in La méthode de Neumann et le problème de Dirichlet (1897). You can find the solution explained in section 8.2 of Poincaré's variational problem in potential theory (2007).