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a harmonic function on a graph which is bounded below and nonconstant


There exists no harmonic functionPlanetmathPlanetmath on all of the d-dimensional grid β„€d which is bounded below and nonconstant. This categorises a particular property of the grid; below we see that other graphs can admit such harmonic functions.

Let π’―3=(V3,E3) be a 3-regular tree. Assign β€œlevels” to the vertices of π’―3 as follows: Fix a vertex o∈V3, and let Ο€ be a branch of π’―3 (an infinite simple path) from o. For every vertex v∈V3 of π’―3 there exists a unique shortest path from v to a vertex of Ο€; let β„“(v)=|Ο€| be the length of this path.

Now define f(v)=2-β„“(v)>0. Without loss of generality, note that the three neighbours u1,u2,u3 of v satisfy β„“(u1)=β„“(v)-1 (β€œu1 is the parent of v”), β„“(u2)=β„“(u3)=β„“(v)+1 (β€œu2,u3 are the siblings of v”). And indeed, 13(2β„“(v)-1+2β„“(v)+1+2β„“(v)+1)=2β„“(v).

So f is a positive nonconstant harmonic function on π’―3.

Title a harmonic function on a graph which is bounded below and nonconstant
Canonical name AHarmonicFunctionOnAGraphWhichIsBoundedBelowAndNonconstant
Date of creation 2013-03-22 12:44:26
Last modified on 2013-03-22 12:44:26
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Example
Classification msc 30F15
Classification msc 31C05
Classification msc 31B05
Classification msc 31A05