About green's representation formula for solutions to Poisson's equation In the exposition of Evan's PDE text, theorem 12 in chapter 2 gives a "representation formula" for solutions to Poissons equation:
$$ u(x) = - \int_{\partial U} g(y) \frac{\partial G}{\partial \nu} (x,y) dS(y) + \int_{U}f(y) G(x,y)dy $$
In the subsequent two sections, he derives the Green's function for the sphere and the half space, and gives two theorems proving that that $u$ defined by the above representation formula with the derived Green's functions substituted in for $G$ satisfy Poisson's equation with the given boundary condition.
I am wondering why $u$ defined by the above representation formula is not just proven in general to satisfy Poisson's equation with boundary condition $g$, and it is instead only proven in a couple of simple cases. Is it too difficult to prove this result in full generality?
 A: Evans proves the representation formula in general (see Theorem 12 p. 35). The theorem states that if $u\in C^2(\bar{U})$ solves the boundary value problem and if Green's function exists, then the representation formula holds.
However, at that point in the book the tools for showing the existence of such an $u$ and $G$ are not yet developed. The required tools are introduced in Chapter 6 in Evans book. Basically, they state that if the boundary of your domain $U$ is regular enough, then such a $G$ exists. If additionally $f,g$ also satisfy some requirements (like Hölder continuity) then such a solution $u$ exists. Nonetheless, these tools only give you the general existence of these functions $u,G$. They don't tell you what they actually look like.
Evans constructs the Green function $G$ in the book for two special cases. However, since we don't know at this point that such solutions $u$ exists we can not apply Theorem 12 to show that the representation formula holds. For example at this point in the book we don't know that a $u$ exists such that $\Delta u = 0$ in $\mathbb{R}^n_+$ with $u(x) \rightarrow u(x^0)$ for $x\rightarrow x^0 \in \partial \mathbb{R}^n_+$, hence we can not apply Theorem 12 to get Theorem 14. Thus, we have to actually verify that our constructed $u$ is a solution to our problem.
