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I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the curvature is not necessarily well defined): $$ K\geq \int_B \frac{\nabla \varphi \cdot \nabla \psi}{\sqrt{1+|\nabla \phi|^2}} \geq 0 \text{ for every } \psi \in C^1(B),\psi \geq 0. $$

I want to uniformly approximate this surface with more regular surfaces, at least $C^2$, with non-negative mean curvature. I guess this should be possible and it is natural to consider a mean-curvature flow $(\phi_t)$ with initial condition $\phi_0=\varphi$.

A result of Ecker and Huisken tells us that $\phi_t$ is smooth for small $t>0$ if the initial condition is uniformly Lipschitz. In this case the hypothesis is verified since $\phi_0$ is $C^{1,\alpha}$. The fact that the initial condition has non-negative mean curvature should be preserved under the flow, so $\phi_t$ will have non-negative mean curvature. On the other hand the bound on the curvature of the initial condition should give a finite upper bound for the curvature of $\phi_t$ (for small $t$). We can then deduce that for $t \in [0,T]$ $\phi_t$ is a uniform approximation of $\phi_0$.

All the above should be correct if the initial condition is $C^2$ (so that the curvature is well defined).

Does the above reasoning still apply in the $C^{1,\alpha}$ case when we only have some information on the "weak" curvature? Do you know any references for this case?

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The statement you want (and a bit more I think) can be found in Lemma 3.8 by Metzger--Schulze in their article "No mass drop for mean curvature flow of mean convex hypersurfaces".

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