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:
- Is there any name or related category of this flow?
- 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.
Best Answer
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 .