ap.analysis of pdes - The normal derivative of the Green's function

I was wondering if anything was known about the following:

Let $mathbb{D}^2=lbrace x^2+y^2< 1 rbrace subset mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ of this disk. I.e. here $pin mathbb{D}^2$ and $G(z;p)$ is smooth and harmonic in $bar{mathbb{D}}^2backslash lbrace p rbrace$, vanishes on the boundary and has the property that $H(z;p) =G(z; p)-log |z-p|$ is smooth.

Now consider the set of functions:
begin{equation}
S=lbrace fin C^infty(partial mathbb{D}^2): f= sum_{i=1}^n lambda_i partial_{nu} G(z; p_i), lambda_i in mathbb{R}, p_i in mathbb{D} rbrace
end{equation}
Here $partial_{nu} G(z; p_i)$ is the normal derivative on $partial mathbb{D}^2$.

My question is what can be said about the set $S$? In particular, is there any hope that it is dense in $L^2(partial mathbb{D}^2)$?

Playing around with things all I got was a big mess so any references would be appreciated.