TITLE: Counterexample for the solvability of $-\Delta u = f$ for $f\in C^{0}$
QUESTION [16 upvotes]: I'm learning for an exam and I'm surprised by the following statement that is given without proof or example:
Let $\Omega\subset\mathbb{R}^n$ be open, bounded and connected and let $f\in C^0(\overline{\Omega})$ (i.e. $f$ is continuous up to the boundary of $\Omega$). Then, in general, there is no $u\in C^2(\Omega)\cap C^0(\overline \Omega)$ satisfying $-\Delta u = f$ in $\Omega$ and $u=0$ on $\partial\Omega$.
I thought that there would always be such a solution under the given prerequisites and that one could even write it down in a closed formula using Green's function...
So what am I missing? Are there sets $\Omega$ as above such that no Green's function exists for them? And can anyone give me an example where no $u$ as above exists?
Thank you very much for your help!

REPLY [10 votes]: You can write the solution as the convolution of Green's function with $f$. It will have the correct Laplacian (in some sense), but it will not necessarily have all second-order partial derivatives as continuous functions. The underlying reason is that inverting the Laplacian and then taking two derivatives amounts to applying a couple of Riesz transforms. These are singular integral operators of  Calderón-Zygmund type, so they map $L^p$ into itself for $1<p<\infty$, and also map $C^{k,\alpha}$ into itself as long as $\alpha$ is not an integer. 
At the integer exponents there is trouble, which already manifests itself in Calculus 2: the formula $\int x^p\,dx = \frac{x^{p+1}}{p+1}$ fails when $p=-1$. The power $x^{-1}$ is unique in that its integral is spread out evenly over all scales. This causes trouble in integral estimates: despite having expected control on each individual scale, we end up with a divergent integral anyway.
Other instances of this: Hilbert transform, harmonic conjugates. E.g., the conjugate of a harmonic function that is continuous on closed disk is not necessarily continuous. But the Hölder (and Dini) continuity is preserved under conjugation.