The motivation of the question comes from a geometric problem: can we approximate a $C^{1,\alpha}$ set $\Omega$ with positive curvature (in distributional sense) from inside with $C^2$ sets with positive curvature? (the approximation is in the Hausdorff distance).

Translating the problem into a graph setting we arrive at the following:

Given a $C^{1,\alpha}$ function $f: B_r \subset \Bbb{R}^d \to \Bbb{R}$ such that $\displaystyle\operatorname{div} \left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right)\geq 0$ can we find a $C^2$ function $g: B_r \to \Bbb{R}$ such that $g \leq f$, $\displaystyle\operatorname{div} \left(\frac{\nabla g}{\sqrt{1+|\nabla g|^2}}\right)\geq 0$ and $\|g-f\|_\infty<\varepsilon$ ?

Do you know any references for this type of results?