# σ-finite

## Trajectory of Some Points on Harmonic Functions

In this ${\mathit{Open Problem}}$ catalogue I plan to keep a record of some of the problems I think about. By ${\mathit{Open}}$ I do not indicate that they are open in the real sense, nor do I pretend they have any deep implications. They are just some problems I find interesting and have had no answer up to the point I publish the posts.

Today I was thinking about PDEs and came across this simple property of harmonic functions:

Theorem 1 {Mean Value Property}Suppose ${\Omega}$ is an open set in ${\Re^{2}}$ and let ${\mathit{u}}$ be a function of class ${\mathcal{C}^2}$ with ${\Delta\mathit{u}=0}$ in ${\Omega}$. If the closure of the disc centered at ${(x,y)}$ of radius ${R}$ is contained in ${\Omega}$, then

$\displaystyle \mathit{u}(x,y)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathit{u}(x+r\cos\theta,y+r\sin\theta)d\theta \ \ \ \ \ (1)$

for all ${0\le r\le R}$.

Now suppose ${\mathit{u}}$ is such a function in ${\Re^2}$ and ${r}$ and ${R}$ be as stated in the theorem.

Then given any ${r_0}$, there exists some point, say, ${(x_0,y_0)}$ on the circle centered at ${(x,y)}$ with radius ${r_0}$ such that ${\mathit{u}(x_0,y_0)=\mathit{u}(x,y)}$, followed from ${\mathit{Mean Value Theorem}}$ of integral.

So my question is, what is the trajectory of such points when ${r}$ ranges over all admissible values.

I do not have an answer yet, and maybe it is related to the ${\mathit{Maximal Principle}}$ of PDEs.