Polynomials on the Sierpinski Gasket with Respect to Different Laplacians which are Symmetric and Self-Similar
Christian Loring, W. Jacob Ogden, Ely Sandine, Robert S. Strichartz
TL;DR
This paper explores polynomial solutions related to various symmetric, self-similar Laplacians on the Sierpinski gasket, revealing unexpected differences from the classical Kigami Laplacian case.
Contribution
It introduces a basis of polynomials for these Laplacians, characterizes them by boundary derivatives, and uncovers surprising boundary value relationships and eigenvalue connections.
Findings
Defined monomials with specific boundary derivative properties
Computed monomial boundary values for different Laplacians
Discovered unexpected relationships between monomials and Neumann eigenvalues
Abstract
We study the analogue of polynomials (solutions to for some ) on the Sierpinski gasket () with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on the polynomials with respect to the standard Kigami Laplacian. We define a basis for the space of polynomials, the monomials, characterized by the property that a certain "derivative" is 1 at one of the boundary points, while all other "derivatives" vanish, and we compute the values of the monomials at the boundary points of . We then present some data which suggest surprising relationships between the values of the monomials at the boundary and certain Neumann eigenvalues of the family of symmetric self-similar Laplacians. Surprisingly, the results for the general case are quite different…
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