Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice differentiable?
Here is an explicit counterexample in $B_1$, $$f(x,y) = x\, y \, \log(\log(x^2+y^2))$$ Both $\partial_{xx} f$ and $\partial_{yy} f$ are continuous but $f$ is not twice differentiable at the origin. 


For very general open sets, this is not true. However, assume true if the boundary is sufficiently smooth and the boundary conditions are nice enough (say, Dirichlet). In that case, your question boils down to asking whether a continuous function whose Laplacian is continuous, too, is also in C^2. This is surely true if instead of continuity you assume Hölder continuity: i.e., if your aim is to pass from $\Delta u\in C^{0,\alpha}$ to $u\in C^{2,\alpha}$: this is exactly the assertion of Schauder's estimates; or else if you content yourself with weak differentiability instead of classical one: in that case, see e.g. §6.3.2 in Evans' Partial Differential Equations. 

