If \(f \in C_c^2(\mathbb{R}^d)\) the convolution of the fundamental solution and \(f\), i.e.

\[u(x) = \phi * f(x),\]

solves Poisson’s equation . [1, 2.2 Theorem 1]

The idea is, that according to (0x6811d54f) we are able to apply \(\Delta\) onto \(\phi\). Then, the integral vanishes anywhere and the singularity of \(\phi\) ensures that \(\Delta u(x)=f(x)\). To prove that, we split the integral into two parts.

References Link to heading

1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.