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?



Approximation with a more regular function and an inequality constraint
Looks like the answer to this question is affirmative. In fact the question is equivalent to the following one: [Regularization by mean curvature flow](https://mathoverflow.net/questions/239981/regularization-by-mean-curvature-flow) 


Take a look at the article in the given answer to see the proof.


