In my research I arrived at the following equation: $$ \int_B \frac{\nabla u \cdot \nabla \varphi}{ \sqrt{1+|\nabla u|^2}}=\int_B f \varphi, (*)$$ for every $\varphi \in C^1(B)$, which is a weak form of the mean curvature equation. (Indeed, if $u$ is $C^2$ then we can integrate by parts and get that the curvature of the graph determined by $u$ is $f$.) I kept searching for some regularity results which apply to my setting, but failed to find something which works.
In my context I know that $(*)$ is satisfied, where $B$ is a $d$-dimensional ball, $u$ is of class $C^{1,\alpha}$ with $\alpha \in (0,1)$ and that $f \in C^\beta$ with $\beta \in (0,1)$.
My question is: is there any chance of obtaining that $u$ is $C^{1,1}$ or $C^2$ from these assumptions? Are there any standard references to this type of results?
