Polynomials on the Sierpiński gasket with respect to different Laplacians which are symmetric and self-similar

Abstract

We study the analogue of polynomials (solutions to $\Delta u^{n+1}u=0$ for some $n$) on the Sierpiński gasket (SG) 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 SG. 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 from the results for the Kigami Laplacian.