In physics context, the following integral appears.
$$ \int \limits_{\mathbb{R}^3} \frac{\partial^2}{\partial y_i \partial y_j} \left( \frac{1}{|x-y|} \right) \mathrm{d}y$$
I believe by the intuition that "one can integrate by parts", one concludes that in the physics context, the integral is equal to zero. However, I am not sure how to make sense of it mathematically and whether I should believe the claim.
In the context of distributions, I understand why for all compactly supported test functions $\phi$, the following is defined.
$$ \int \limits_{\mathbb{R}^3} \frac{\partial^2}{\partial y_i \partial y_j} \left( \frac{1}{|x-y|} \right) \phi(y) \mathrm{d}y := \int \limits_{\mathbb{R}^3} \frac{1}{|x-y|} \partial_i \partial_j \phi(y) \mathrm{d}y$$
Now, my idea was to consider a sequency of compactly supported functions $\phi_n$ such that as $n \to \infty$, they converge pointwise to a function that is $1$ everywhere. Then, I would interpret the first integral in the question as the following limit.
$$ \int \limits_{\mathbb{R}^3} \frac{\partial^2}{\partial y_i \partial y_j} \left( \frac{1}{|x-y|} \right) \mathrm{d}y := \lim \limits_{n \to \infty} \int \limits_{\mathbb{R}^3} \frac{1}{|x-y|} \partial_i \partial_j \phi_n(y) \mathrm{d}y $$
However, if I, for example, take $\phi_1$ to be such that it is $1$ for all $|y| < 1$, then I can define $\phi_n(y) = \phi_1(y/n)$. Then, $\phi_n$ is still compactly supported and it is $1$ for all $|y| < n$. So, as $n \to \infty$, $\phi_n$ tends to a function that is $1$ everywhere. However, from the scaling property I can also compute that $\partial_i \partial_j \phi_n(y) = 1/n^2 \cdot \partial_i \partial_j \phi_1(y/n)$.
Then, I would get by change of variables $u = y/n$, the following. $$\lim \limits_{n \to \infty} \int \limits_{\mathbb{R}^3} \frac{1}{|x-y|} \partial_i \partial_j \phi_n(y) \mathrm{d}y = \lim \limits_{n \to \infty} \int \limits_{\mathbb{R}^3} \frac{n}{|x-nu|} \partial_i \partial_j \phi_1(u) \mathrm{d}u$$
Then, if we consider the special case of $x = 0$, I would get the following. $$\lim \limits_{n \to \infty} \int \limits_{\mathbb{R}^3} \frac{n}{|nu|} \partial_i \partial_j \phi_1(u) \mathrm{d}u = \int \limits_{\mathbb{R}^3} \frac{1}{|u|} \partial_i \partial_j \phi_1(u) \mathrm{d}u \neq 0 $$
So, is there any way to make sense of the first integral, and can one somehow argue that it is zero?