The definition of generalized mean curvature on $C^1$ hypersurfaces is given as follows:

Let $M$ be a closed orientable $C^1$ hypersurface in $\mathbb{R}^{n+1}$ and $\mu$ be the $n$-dimensional Hausdorff measure. We say the $\mathbb{R}^{n+1}$ valued vector function $\textbf{H}$ is a generalized mean curvature on $M$, if for any smooth vector fields $X \in \mathbb{R}^{n+1}$, the following identity holds: $$\int_{M} div_MX d\mu= \int_{M} X \cdot \textbf{H} d\mu,$$where $\dim_MX$ is the tangential gradient, that is, if $e_1, \cdots, e_n$ form an orthomormal tangent bundle on $M$, then $div_M X= \sum_{i=1}^n\nabla_{e_i} X \cdot e_i$.

My question is, is there a concrete example that $M$ is only $C^1$ and the generalized mean curvature can be constructed?

This question has constantly intrigued me for a long time. The first time when I was interested in this question was when I was reading the paper by Micheal and Simon http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160260305/abstract. In the paper, the authors introduced the generalized mean curvature possibly defined on very rough manifolds, and then they proved the generalized Micheal-Simon Sobolev inequality. Later on I constantly occured papers with the setting that the initial surface is just rectifiable and has bounded generalized mean curvature. I just don't want things to be pused into the most general setting.

As far as I know, even given a $C^{1,\alpha}$ manifold, the generalized mean curvature can only be defined as a measure, and cannot be written as a vector valued function, see https://eudml.org/doc/252366. Can anyone give me some concrete nontrivial examples, and why people study flows with such weak assumption? For example, Huisken and Ilmanen deals with inverse mean curvature flow on $C^1$ manifolds with bounded mean curvature, see https://projecteuclid.org/euclid.jdg/1226090483. I've no idea what is the motivation...

I'll really appreciate if anyone can share any ideas. Thanks!

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