2
$\begingroup$

Possible Duplicate:
A stronger version of discrete “Liouville’s theorem”

Let each lattice point of the plane be labeled by a positive real number . Each of these numbers is the arithmetic mean of its four neighbors ( above , below , left , right ) . Then is it true that all the labels are equal ? ( I have only been able to prove the equality of all labels when all the labels are positive integers but can not seem to get a hold when the labels are arbitrary positive reals )

  • 0
    Anyway, if you drop the positivity restriction, at integer lattice points take $f(x,y) = xy,$ this satisfies your condition. So there is a relationship with smooth harmonic fubctions, it is just not clear how far the analogy goes. Most of what I see on graph Laplacians is about finite graphs.2012-10-30

0 Answers 0