It is well known that \begin{equation} \Delta_1\frac{1}{|\vec{r}_1-\vec{r}_2|}=-4\pi\delta(\vec{r}_1-\vec{r}_2), \end{equation} where the index 1 in $\Delta_1$ denotes differentiation with respect to $\vec{r}_1$. One way to see this is for example by introducing a regulator \begin{equation} \frac{1}{|\vec{r}_1-\vec{r}_2|}\rightarrow\lim_{a\rightarrow 0}\frac{1}{\sqrt{(\vec{r}_1-\vec{r}_2)^2+a^2}} \end{equation} and performing the integration over a sphere. This is probably not a proper proof, but that's a way often used by physicists and rigorous enough for me.
I was wondering if there is also a relation like \begin{equation} \nabla_1\cdot\nabla_2\frac{1}{|\vec{r}_1-\vec{r}_2|}=4\pi\delta(\vec{r}_1-\vec{r}_2), \end{equation} and if so, how one would show or disprove it. I tried the way outlined above, but this is in the second case much more complicated. Or is it just trivially zero? (as it should be the case for $|\vec{r}_1-\vec{r}_2|\neq 0$)