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 $\lambda_k=\frac{-1}{2k+1}$ with multiplicity $2k+1$, $k=0,1,...$.

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.

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.