Inhomogeneous Laplace equation

Question

It is well known that the Laplace equation $$\Delta f=0$$ has many solutions in $\mathbb{R}^2$, but what about the inhomogeneous Laplace equation $$\Delta f=g$$ Can anyone give me a reference which discuss this equation? I want to know the condition on $g$ which makes this equation solvable, in particular, I want to know if the equation is solvable for any smooth $g$.

Answer 1

As a starting point, assume that $f \in C_c^2(\mathbb{R}^2)$, let $\Phi(x) = -\frac{1}{2\pi}\log|x|$ and set $$ u(x) = \int_{\mathbb{R^2}}\Phi(x - y)f(y)\,dy. $$ Then one has the following result

Theorem: Let $u$ be as above, then $u \in C^2(\mathbb{R}^2)$ and $-\Delta u = f$ in $\mathbb{R}^2$.

You can find a proof of this theorem in Evans' PDE book. The Wikipedia page about Poisson's equation is also a good read if you are relatively new to this kind of problems.