The classical Schauder estimates (see the link)
<http://en.wikipedia.org/wiki/Schauder_estimates>


Requires $f\in C^\alpha$ in order to get a solution $u\in C^{2+\alpha}$ of the equation


$$\Delta u=f$$


In fact, we can construct a continuous function f, which is not Hölder of any order on a positive meausre set. 


Is it possible to find solution $u\in C^2$?


Or is there any counterexample to show such solution $u$ does not exist？



The Hölder continuity condition of the Schauder estimates
Merely $f\in C$ cannot guarantee $u\in C^2$, as there are examples of $u\not\in C^2$ with $\Delta u\in C$, cf. an exercise in Gilbarg-Trudinger. This failure can be formulated in terms of non-closedness of the range of the Laplacian, or unboundedness of the inverse between certain spaces. The spaces $C^k$ (as well as $C^{k,1}$) are not well suited for studying elliptic partial differential equations. This is essentially the reason why we use the Holder and Sobolev spaces to study them.


