Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

Question

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 $$\frac{C_2}{|x|^{n-2}} \le u(x) \le \frac{C_1}{|x|^{n-2}}.$$

Answer 1

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

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 (1963), "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.

Answer 2

the result is in te classical paper Regular points for elliptic equations with discontinuous coefficients by LITTMAN, STAMPACCHIA and WEINBERGER.

Another source more easy to read is in the paper "The Green function for uniformly elliptic equations" by Gruter and Widman (1982) published in Manuscripta Mathematica. This is quite self contained and more accessible.