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.



Eigenvalues of the double layer potential
This is a classic problem solved by Poincaré in [La méthode de Neumann et le problème de Dirichlet](http://link.springer.com/article/10.1007/BF02418028) (1897). You can find the solution explained in section 8.2 of [Poincaré's variational problem in potential theory](http://www.math.ucsb.edu/~mputinar/poincare.pdf) (2007).


