Higher order Laplacians on p.c.f. fractals with three boundary points and dihedral symmetry
Volume 257 / 2021
Abstract
We study higher order tangents and higher order Laplacians on fully symmetric p.c.f. self-similar sets {with three boundary points}. Firstly, we prove that for any function $f$ defined near a vertex $x$, the higher order weak tangent of $f$ at $x$, if exists, is the uniform limit of local multiharmonic functions that agree with $f$ near $x$ in some sense. Secondly, we prove that the higher order Laplacian on a fractal can be expressible as a renormalized uniform limit of higher order graph Laplacians. Some results can be extended to general p.c.f. self-similar sets. In the Appendix, we provide a recursive algorithm for the exact calculations of the boundary values of the monomials on $D3$ symmetric fractals, which is shorter and more direct than in the previous work on the Sierpiński gasket.
