I have two questions about the analyticity of solutions of Poisson equation: Let $f$ be a real-valued smooth function on $\Omega \subset \mathbb{R}^2$, as we know that the Poisson equation $$\Delta u = f $$ can have many solutions (up to harmonic functions). I wonder that
- If $f$ is analytic, is there an analytic solution for the Poisson equation in general? The reference to this question is welcome!
- Is there an example (or proof) that if $f$ is not analytic then its solutions are not analytic?