In the well known paper by LittmanWeinbergerStampacchia "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 $$ \frac{C_1}{x^{n2}}\leq \Gamma(x)\leq \frac{C_2}{x^{n2}}\quad\mbox{in }\mathbb{R}^n, $$ 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 $$ \nabla \Gamma(x)\leq \frac{C_3}{x^{n1}}\quad\mbox{in }\mathbb{R}^n, $$ 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 $$ \nabla G(x)\leq \frac{C_3}{x^{n1}}\quad\mbox{for }x \in B_1. $$ Otherwise, is there a reference where one could find the proof of this last result?
1 Answer
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

$\begingroup$ If I remember correctly in Thm. 3.3. they proved pointwise estimates for the gradient on general open domain with costant depending also on $n, \lambda, L, \Omega$ and the modulus of continuity. Later, in Thm. 3.4 they proved that if $\Omega$ is convex then the estimate depends on $n, \lambda, L,$ the modulus of continuity of the coefficient and the diameter of $\Omega$. I was wondering if in the case $\partial \Omega$ smooth and smooth coefficient the estimates were independt on $\mathrm{diam}(\Omega)$. $\endgroup$– Gio712Dec 2, 2021 at 9:45

$\begingroup$ I checked for the explicit expression of the Green function in the case of Laplace operator in the ball centered at the origin having radius R. I found the explicit expression at page 20 of GilbargTrudinger. The number R is present so I believe the constant does depend on R. $\endgroup$ Dec 2, 2021 at 10:15

$\begingroup$ I have the feeling that in many cases gradient estimates can follow from those of the fundamental solution itself, at least in the whole space. To explain, let us consider with the parabolic fundamental solution $p(t,x,y)$; assuming upper Gaussian estimates, that for $\Gamma$ follows by integrating in time ($n \geq 3$) and that for the gradient if one can prove gaussian estimates for $\sqrt {t} p_x$. A way for proving these last is to show first that $tp_t$ satisfies Gaussian estimates (by analyticity in time), then $t Lp$ by difference ($L$ is the operator) and then try to interpolate. $\endgroup$ Dec 2, 2021 at 10:55