Let $u$ be an harmonic function in a cylindrical domain $B\_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma\_t=\{u=t\}$ are graphs of functions on $B\_2^{n-1}$.
Consider a linear parametrization of $u$:
$$u\_t:=u-t.$$
Then the nodal set of $u\_t$ is $t$-level set of $u$:
$$\{u\_t=0\}=\{u=t\}.$$
We can regard this $u\_t$ to be the level set function of evolving surface $\Gamma\_t=\{u=t\}$. If we denote $v$ as the vector field indicating the speed of the flow of $\Gamma\_t$ follows, then by level set equation
$$|v|=\frac{1}{|\nabla u|}.$$
Since we know that $v$ is normal to $\Gamma\_t$, $v$ is parallel to $\nabla u$, hence
$$
 v=\frac{\nabla u}{|\nabla u|^2}.
$$
On the other hand, since $u$ is constant on its level surfaces $\Gamma\_t$, its Laplace-Beltrami operators are identically zero:
$$0=\Delta\_{\Gamma\_t}u=\Delta u+H\_{\Gamma\_t}\partial\_{\nu}u-\partial\_{\nu\nu}u=H\_{\Gamma\_t}\nabla u-\partial\_{\nu\nu}u.$$
Therefore with the above expression, we obtain
$$
 v=\frac{|H\_{\Gamma\_t}|}{|\partial\_{\nu\nu}u|}\nu,
$$
where $\nu:=\nabla u/|\nabla u|$ is a unit normal vector to $\Gamma\_t$.


So we have that, the surface $\Gamma\_t$ follows some geometric flow somehow related to mean curvature flow (if $|\partial\_{\nu\nu} u|=1$ then it is a mean curvature flow). My questions are:


1. Is there any name or related category of this flow?
2. I am interested in the $C^{2,\alpha}$ regularity of $\Gamma\_t$, if $\Gamma\_t$ was a graph of $C^{2,\alpha}$ function, then will $\Gamma\_s$ be $C^{2,\alpha}$ in some neighborhood of $t$? If so, can we control $C^{2,\alpha}$ norm of $\Gamma\_s$ by $\Gamma\_t$?


Just to note, different formulation of the curvature equation above ($0=H\_{\Gamma\_t}\nabla u-\partial\_{\nu\nu}u$) is that
$$|\nabla u|\Delta\_1 u=\Delta\_\infty u,$$
where $\Delta\_p$ is $p$-Laplacian. It is interesting for me that two operators with duality have some relation. Unfortunately I don't have much knowledge in this direction ($p$-laplacian) so I could not have any meaningful result.



Geometric flow by the level sets of a harmonic function
There is a lot of current study of the level sets of harmonic functions in this exact way.


See these papers: <https://arxiv.org/abs/1209.4669> <https://arxiv.org/abs/2108.08402> and <https://arxiv.org/abs/1911.06754> .


Some may not look directly related to what you're asking, but note that anytime you are using the co-area formula, it's like integrating $\frac{d}{dt} \int\_{u=t} F$ with respect to $t$, so it ends up being related to what you're asking.


In terms of the $p$-Laplacian, there is some relationship between your flow and inverse mean curvature flow. See <https://arxiv.org/abs/1812.05022> .


