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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 )

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    This might be true, I'm not sure. The mean value property you describe is true for harmonic functions in the plane, where the mean value is given by an integral. In that case, Liouville's theorem says that any bounded harmonic function is constant. It is not, however, clear to me that this is guaranteed to hold in the discrete case. Where did you get this?2012-10-30
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    http://en.wikipedia.org/wiki/Five-point_stencil#Two_dimensions2012-10-30
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    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

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