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arXiv · 1901.08713

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
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Abstract

We study the analogue of polynomials (solutions to $\Delta^{n+1} u =0$ for some $n$) on the Sierpinski 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.

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Christian Loring, W. Jacob Ogden, Ely Sandine, Robert S. Strichartz. 2019-01-25. Polynomials on the Sierpinski Gasket with Respect to Different Laplacians which are Symmetric and Self-Similar. https://arxiv.org/abs/1901.08713

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