PEnergy on the Sierpinski Gasket

My name is Eddie Herman and Professor Strichartz
and I studied Penergy on the Sierpinski Gasket. The case when P=2 is mostly
understood. We know that given any boundary values on the gasket, 2energy
on any other level with respect to these boundary values is 3/5 the previous
level. The value 3/5 for P=2 is called the r_p value. One of the goals
this summer was to find the r_p value for any P. In other words, we want
to find a function that given P gives r_p.

Another goal was to find the pth harmonic extension
algorithm. For p=2 this algorithm is very nice as it is, given a vertex,
2/5 the nearest neigbbor plus 2/5 the other nearest neigbor, plus 1/5 the
furthest boundary value. For general p, however, the algorithm is nonlinear
and much more complicated. Some of these algorithms for different pvalues
are below in the data. The p's are situated in terms of their relationship
with the number q such that 1/p +1/q = 1.

Here is a graph of
different penergy functions on the same graph.

Here is a graph of
pvalues and there corresponding r_p values.

Here is a graph
function approximating the r_p function.
This is the code for finding PEnergy on the Gasket
(It seems maple
display does not come out well in html file so here is plain text)
This is Data for different Pvalues
with the Code above

Additions, Corrections or Feedback
on this page to:
peherman@midway.uchicago.edu
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Last Update: September 21, 1999