Green function of the Laplace-Dirichlet operator in a bounded domain Let $\Omega \subset \mathbb{R}^d$, $d = 1,2,3$, be a bounded domain with $\partial \Omega$ smooth. Let $$A = -\Delta \colon H_0^1(\Omega) \cap H^2(\Omega) \to L^2(\Omega)$$be the Laplace-Dirichlet operator. I am looking for a reference that assures the existence of a Green operator for $A$, namely $G \colon L^2(\Omega) \to H_0^1(\Omega) \cap H^2(\Omega)$ such that $$v = Gf \Longleftrightarrow Av = f.$$
In addition, what do we know about it? I am particularly interested in compactness of this Green operator
 A: Consider the problem of find $u\in H_0^1$ such that $$\tag{1}\int_\Omega\nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in H_0^1$$
There is a lot of ways to solve this problems, to wit, one can use Lax-Milgram theorem or Calculus of Variation or Monotone Methods operators and so on. Let's try the second one (because I really like it more than others).
Define $I:H_0^1\to\mathbb{R}$ by $$I(u)=\frac{1}{2}\int_\Omega |\nabla u|^2-\int_\Omega fu$$
Note that $I$ is the energy functional associated to the problem $(1)$, i.e. $$\langle I'(u),v\rangle = \int_\Omega\nabla u\nabla v-\int_\Omega fv, \forall\ u,v\in H_0^1$$
where $\langle \cdot,\cdot\rangle$ denotes duality between $H_0^1$ and $H^{-1}$. Hence, to solve $(1)$ is equivalently to find $u\in H_0^1$ such that $$\langle I'(u),v\rangle =0,\ \forall\ v\in H_0^1$$
The last equality is saying that $u$ is a critical point of $I$. Hence, let's try to find a critical point of $I$ (in fact a minimum point).
Note that (by using Holder and Poincare inequality) there exist a constant $C>0$ such that $$\tag{2}I(u)\geq \frac{1}{2}\|u\|_{1,2}^2-C\|u\|_{1,2}$$
From $(2)$ we conclude that $I$ is coercive ($I(u)\to\infty$ if $\|u\|_{1,2}\to\infty$). Now let's prove that $I$ is lower semicontinuous in the weak topology. Let $u_n\to u$ in the weak topology. On one hand we have that $\|\cdot\|_{1,2}$ is lower semicontinuous in the weak topology. On other hand, note that by the compact immersion of $H_0^1$ in $L^2$, we can suppose that (up to a subsequence) $u_n\to u$ strongly in $L^2$, hence, by using Lebesgue theorem, we can conclude that $\int_\Omega fu_n\to \int_\Omega fu$. Therefore $I$ is weak lower semicontinuous.
By using the fact that $I$ is coercive and weak lower semicontinuous, we can conclude that there exist $u\in H_0^1$ such that $I(u)\leq I(v)$ for every $v\in H_0^1$. Because $I$ is strictly convex, we also have that $u$ is unique, hence we can associate to $(1)$ the Solution operator or Green operator $G$, i.e. if $G(f)=u$ then $u$ satisfies $(1)$. Note that by regularizing the solution (see Brezis chapter 9) we can suppose that $u\in H_0^1\cap H^2$, therefore $$G:L^2\to H_0^1\cap H^2$$
The first thinkg we can cnote about $G$ is it's continuity: indeed, by taking $v=u=G(f)$ in $(1)$ we have by Holder and Poincare inequality that $$\|G(f)\|_{1,2}\leq C\|f\|_2$$
On the other hand, because $H_0^1\cap H^2$ is compactly embedded in $L^2$, we can conclude that $$G:L^2\to L^2$$
is a compact operator. 
Remark: You can also show that $G$ is positive and symmetric.
Remark 1: All that I have said here can be found for example in: Brezis chapter 9, Evans chapter 6 and so on...
