P-Energy on the Sierpinski Gasket

    My name is Eddie Herman and Professor Strichartz and I studied P-energy on the Sierpinski Gasket. The case when P=2 is mostly understood. We know that given any boundary values on the gasket, 2-energy 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 p-th 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 non-linear and much more complicated. Some of these algorithms for different p-values 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 p-energy functions on the same graph.
    Here is a graph of p-values and there corresponding r_p values.
    Here is a  graph  function approximating the r_p function.

This is the code for finding P-Energy 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 P-values 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