a harmonic function on a graph which is bounded below and nonconstant
There exists no harmonic function 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 βlevelsβ to the vertices of π―3 as follows: Fix a vertex oβV3, and let Ο be a branch of π―3 (an infinite simple path) from o. For every vertex vβV3 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 (βu1 is the parent of vβ), β(u2)=β(u3)=β(v)+1 (βu2,u3 are the siblings of vβ). 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 |
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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 |