TITLE: On the fundamental solution for elliptic PDE
QUESTION [5 upvotes]: In the well known paper by Littman-Weinberger-Stampacchia "Regular points for elliptic equations with discontinuous coefficients", the authors were able to prove the validity the following statement: given a bounded, measurable and uniformly elliptic matrix $A$ (roughly speaking $\lambda\,\mathrm{Id}\leq  A \leq L \,\mathrm{Id}$, for some $\lambda \leq L$) and $\Gamma$ the fundamental solution in $\mathbb{R}^n$, that is $\Gamma\geq 0$ and
$$\label{ciap}
-\mathrm{div}(A\nabla \Gamma)=\delta_0 \quad\mbox{in }\mathbb{R}^n
$$
then
$$
\frac{C_1}{|x|^{n-2}}\leq \Gamma(x)\leq \frac{C_2}{|x|^{n-2}}\quad\mbox{in }\mathbb{R}^n,
$$
for some $C_1,C_2$ depending only on $n,\lambda$ and $L$. I was wondering if it is true that, if $A$ is smooth, then
$$
|\nabla \Gamma|(x)\leq \frac{C_3}{|x|^{n-1}}\quad\mbox{in }\mathbb{R}^n,
$$
for some $C_3$ depending only on $n,\lambda$ and $L$.
Indeed, in the book "Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems" by Kenig, he states in Theorem 1.2.8 that a similar estimate holds for the Green function of operator with smooth coefficient $A$ in smooth domain $\Omega$ for some constant $C_3$ depending only on $n, \lambda $ and $L$ (he does not mention the possible dependence from the diameter of $\Omega$ but the proof is omitted): if $-\mathrm{div}(A\nabla G)=\delta_0$ in $B_1$ and $G=0$ on $\partial B_1$ then
$$
|\nabla G|(x)\leq \frac{C_3}{|x|^{n-1}}\quad\mbox{for }x \in B_1.
$$
Otherwise, is there a reference where one could find the proof of this last result?

REPLY [5 votes]: In the paper The Green Function for uniformly elliptic equations Manuscripta Mathematica Gruter & Widman proved - among other things - pointwise estimates about the gradient of the Green function under the Dini continuity assumption on the leading coefficients. This is in section 3 Theorem 3.3