# σ-finite

Things add up.

## Category Archives: Open Problems

## Trajectory of Some Points on Harmonic Functions

December 6, 2010

Posted by on In this catalogue I plan to keep a record of some of the problems I think about. By 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 is an open set in and let be a function of class with in . If the closure of the disc centered at of radius is contained in , then

for all .

Now suppose is such a function in and and be as stated in the theorem.

Then given any , there exists some point, say, on the circle centered at with radius such that , followed from of integral.

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

I do not have an answer yet, and maybe it is related to the of PDEs.