From the Green's function definition, it seems we only require $G(x,y)\in C(\bar{U}) \cap C^2(U)$.
In the theorem 12, we have a term $\frac{\partial G}{\partial v}(x,y)$. Since it is a directional derivative on the boundary and we have used Green's theorem ealier on . Do we need $G(x,y) \in C^2(\bar{U})$?