Perhaps the best self contained reference on this result is the book [2] by Stampacchia himself. It is a set of typewritten course notes in French, taken from a graduate course on elliptic equations held by Stampacchia at the Centre de Recherches Mathématiques of the Montreal University in the summer of 1965. As a course, it starts by introducing basic concepts and definitions from the theory of Sobolev spaces, elliptic operators and goes on up to touching general nonlinear problems. Similarly to what is done in [1] (§7, theorem 7.1 p. 66), the result is proved as a corollary ([2], ch. 8, p. 235) of a general **theorem** ([2], ch. 8, théorème 8.5, pp. 234-235): *let $\mathscr{G}$ and $\mathscr{\bar G}$ be Green's functions of the following problems*
$$
\begin{cases}
-\mathrm{div}\big(A(x)\nabla\mathscr{G}(x,y)\big)=\delta(x-y)\\
\left.\mathscr{G}\right|_{x\in\partial\Omega}=0
\end{cases}\quad
\begin{cases}
-\mathrm{div}\big(\bar{A}(x)\nabla\mathscr{\bar{G}}(x,y)\big)=\delta(x-y)\\
\left.\mathscr{\bar{G}}\right|_{x\in\partial\Omega}=0
\end{cases}
$$
*where $A(x)$ and $\bar{A}(x)$ are matrices with bounded measurable coefficients and the same ellipticity constant $C$*, i.e.
$$
\begin{split}
C^{-1}\mathrm{Id} \le A(x) & \le C \mathrm{Id}\\
C^{-1}\mathrm{Id} \le \bar{A}(x) & \le C \mathrm{Id}
\end{split}
$$
*Then the following estimate holds*
$$
K^{-1}\le \frac{\mathscr{G}(x,y)}{\mathscr{\bar G}(x,y)}\le K\qquad \forall x,y\in\Omega^\prime
$$
where

- $K=K(C,\Omega,\Omega^\prime,n)$ is a positive constant
- $\Omega^\prime\Subset\Omega$ is any compact subset of the open domain $\Omega\Subset\mathbb{R}^n$.

Assuming $\bar{A}(x)\equiv\mathrm{Id}$, $\mathscr{\bar G}(x,y)$ becomes the Green's function for the Laplace operator in the domain $\Omega$ and the sought for estimate is an easy consequence of the theorem.

**Three observations**

- The notation used in [2] is updated respect to the one used in [1]: for example the ball is not indicated with $\Sigma$ but with a perhaps more comprehensible $I(x,R)$. In sum the text notation is closer to the current standards respect to [1].
- In [2] it is clearly stated (and proved) that
*the above estimate holds on every compact $\Omega^\prime\Subset\Omega$, not only for balls*.
- Stampacchia
*assumes that $\Omega\Subset\mathbb{R}^n$ is $\mathrm{H}^1_0$-admissible* ([2], ch. 8, p. 217):

**References**

[1] Walter Littman, Hans Weinberger and Guido Stampacchia (1962), "Regular points for elliptic equations with discontinuous coefficients", Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, serie III, Vol. 17, n° 1-2, pp. 43-77, MR161019, Zbl 0116.30302.

[2] Guido Stampacchia (1966), "*Équations elliptiques du second ordre à coefficients discontinus*" (notes du cours donné à la 4me session du Séminaire de mathématiques supérieures de l'Université de Montréal, tenue l'été 1965), (in French), Séminaire de mathématiques supérieures 16, Montréal: Les Presses de l'Université de Montréal, pp. 326, ISBN 0-8405-0052-1, MR0251373, Zbl 0151.15501.