Justification of use of delta functions/rigorous proof of Green's function for Poisson equation

I'm looking at the proofs of Helmholtz's theorem, but I'm having trouble justifying their interchange of integration and differentiation and subsequent treatment of the "integrand" as a delta function.

Referring to http://www.ph.ed.ac.uk/~rhorsley/SI10-11_t+f/lec19.pdf, on page 2, this essentially boils down to showing that $v(r) = -\frac{1}{4\pi } \int \frac{a_i(r')}{|r-r'|} dr'$ solves the Poisson equation $\nabla ^2 v_i = a_i$, ie. showing that $-\frac{1}{4\pi |r-r'|}$ is the Green's function for the Poisson equation. This seems somewhat familiar to me but I'm not sure how to prove it rigorously. The proofs I've seen of this again use Dirac delta arguments which again seem a bit hand-wavey.

Could anyone help me sure up my logical understanding or direct me somewhere with a more careful treatment please?

EDIT: Hmm, I found a proof for the validity of the Green's function that looks convincing at quick glance (Theorem 1.1 of Week 7 from http://math.mit.edu/~jspeck/18.152_Fall2011/18152_Schedule.html ).

• Jared's lecture notes give the complete derivation in three spatial dimensions. Most partial differential equations textbooks which covers elliptic PDEs would contain this either as a theorem or as a guided homework. It is in, for example, chapter two of Gilbarg and Trudinger's Elliptic partial differential equations of second order, and in chapter two of Evans' Partial Differential Equations. – Willie Wong Feb 13 '13 at 13:55

I found the utilization of a Fourier Transform the most convincing argument. I.e. transforming the PDE gives $$\mathscr{F}\left\{-\nabla^2 v(x) = a(x)\right\} \equiv |\omega|^2 V(\omega) = A(\omega) \Leftrightarrow V(\omega) = \frac{1}{|\omega|^2}A(\omega) \Leftrightarrow v(x) = \mathscr{F}^{-1}\left\{\frac{1}{|\omega|^2}\right\}\ast a(x)$$ Since $\mathscr{F}^{-1}\{1\}=\delta(x)$, it can be said the Fundamental solution solves $-\nabla^2 v(x) = \delta(x)$. Note that $x$ and $\omega$ are vectors in $\mathbb{R}^n$. This line of argumentation was taken from Kythe, P. K. Fundamental Solutions for Differential Operators and Applications Birkäuser, 1996 There you'll find a rigorous justification for that approach on the bases of Distributions and a convolution algebra.