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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.

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  • $\begingroup$ Although studying the MCF via level sets has a rich history, the approach chosen there---as far as I know---basically goes in the opposite direction of what you're doing here. I'm not an expert in these questions, but I'm a bit skeptical about the point of view you've chosen. For example, as $\partial_{\nu \nu} u \neq 0$ is needed to define your equation, wouldn't you get the answer to the second question from the implicit function theorem anyway? $\endgroup$
    – Leo Moos
    Oct 22 '21 at 10:23
  • $\begingroup$ @LeoMoos As you can see, if $\partial_{\nu\nu}u=0$ then at the point $H_{\Gamma_t}=0$ as well (here let's assume $\nabla u\neq 0$). In that case I can simply write $v=\nabla u/|\nabla u|^2$. Obtaining $C^{2,\alpha}$ regularity (or more) is easy, but I am interested in getting $C^{2,\alpha}$ estimation of $\Gamma_s$ in terms of $\Gamma_t$, e.g. $\|f_s\|_{C^{2,\alpha}}\lesssim \|f_t\|_{C^{2,\alpha}}$. Thank you for the comment. $\endgroup$ Oct 22 '21 at 11:12
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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 .

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  • $\begingroup$ Great! Thank you for the references, I will take a closer look. $\endgroup$ Oct 23 '21 at 2:27

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