Where can I find a (readable and self-contained) proof of the following result?



> 
> Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\mathrm{div}(A(x)\nabla u) = \delta\_0,$$
> $$u|\_{\partial \Omega} = 0,$$
>  where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$). Then on $B\_{1/2}$, we have 
>  $$\frac{C\_2}{|x|^{n-2}} \le u(x) \le \frac{C\_1}{|x|^{n-2}}.$$
> 
> 
> 



Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs
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](http://www.crm.umontreal.ca) 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**


1. 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].
2. In [2] it is clearly stated (and proved) that *the above estimate holds on every compact $\Omega^\prime\Subset\Omega$, not only for balls*.
3. 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](http://www.numdam.org/item/ASNSP_1963_3_17_1-2_43_0)", Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, serie III, Vol. 17, n° 1-2, pp. 43-77, [MR161019](http://www.ams.org/mathscinet-getitem?mr=161019), [Zbl 0116.30302](https://zbmath.org/?q=an:0116.30302).


[2] Guido Stampacchia (1966), "*[Équations elliptiques du second ordre à coefficients discontinus](http://www.worldcat.org/title/equations-elliptiques-du-second-ordre-a-coefficients-discontinus/oclc/1007223)*" (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](http://www.crm.umontreal.ca/pub/Ventes/Catalogue_SMS_Eng.html), Montréal: Les Presses de l'Université de Montréal, pp. 326, ISBN 0-8405-0052-1, [MR0251373](https://www.ams.org/mathscinet-getitem?mr=0251373), [Zbl 0151.15501](https://zbmath.org/?q=an:0151.15501).


