Question

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？

Answer 1

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.

Answer 2

For $u$ to be from $C^2$ it is enough that the modulus of continuity of $f$ satisfies the Dini condition. For example, modulus of continuity $\omega(h)=1/(|log h|+1)^2$ is not marjorised by the Holder modulus of continuity $\omega(h)=C h^\alpha$.

The same goes for coeffitients of elliptic equations. In the work “A priori bounds and some properties for solutions of elliptic and parabolic equations”, Uspekhi Mat. Nauk, 18:4(112) (1963), 215–216, Kruzhkov notes that the problem of continuity of higher derivatives of solutions of elliptic equations is closely connected with the question of when for a function of several independent variables condition of the existence of pure continuous derivatives of order net $l\ge2$ implies the existence of mixed derivatives of the same order. He gives nesessary and suffitient condition for it. Namely, denote $\omega(h)$ the modulus of continuity of the order $l$ pure derivatives of $u$ and $\omega^*(h)=\int_0^h\frac{\omega(t)}t dt$ the second modulus of continuity. Then in order that all mixed derivatives were continuous (in some domain $Q\subset\mathbb R^n$) it is necessary and sufficient that the module continuity $\omega^* (h) $ satisfies the Dini condition. Moreover, if $\omega^* $ satisfies the Dini condition, then all mixed derivatives of order $l$ in any strictly interior subdomain $Q'\subset Q$ have modulus of continuity, which does not exceed $Cw^* $. If $\omega^*$ does not satisfy the Dini condition, then there exists a function $u$ which at some point of $Q$ have no mixed derivatives of order $l$.