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

• HTML for the P-energy polynomial finder
• Maple Code for above HTML
• Here is Professor Back's bisect method needed in my program
• Maple Code for above HTML
•  Here is Melanie Pivarski's decimate program needed in my program
•  Maple Code for above HTML

This is Data for different P-values with the Code above

•  Data for p=8/3 and p=8/5, respectively
•  Maple Text for this data
•  Data for p=20/9 and p=20/11, respectively
•  Maple text for this data
•  16/9 and 16/7,respectively
•  Maple for above
•  4/3 and 4,respectively
•  Maple for above
•  10 and 10/9,respectively
•  Maple for above
•  6 and 3 data,respectively
•  Maple for above

Additions, Corrections or Feedback on this page to:

peherman@midway.uchicago.edu

Last Update: September 21, 1999