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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?

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This question is solved in the PhD thesis of Nicolas Landais: Problèmes de régularité en optimisation de forme. You can read the thesis here. The result is presented in Chapter 6. The conclusion is that with the given hypotheses the function $u$ is $C^{2,\alpha}(B)$. Nicolas Landais also has two publications which are related to this subject.

The proof uses some results from the book of Gilbarg and Trudinger.

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