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Tag Archives: Mean Value Property

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.

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