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Higher order Laplacians on p.c.f. fractals with three boundary points and dihedral symmetry

Volume 257 / 2021

Shiping Cao, Hua Qiu Studia Mathematica 257 (2021), 313-345 MSC: Primary 28A80; Secondary 31C25. DOI: 10.4064/sm191017-24-5 Published online: 12 October 2020


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.


  • Shiping CaoDepartment of Mathematics
    Cornell University
    Ithaca, NY 14853, U.S.A.
  • Hua QiuDepartment of Mathematics
    Nanjing University
    Nanjing 210093, P.R. China

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