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p-Energy and p-Harmonic Functions on Sierpinski Gasket Type Fractals

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Abstract

We show that it is possible to define a notion of p-energy for functions defined on a class of fractals including the Sierpinski gasket (SG) for any value of p, 1<p<∞, extending the construction of Kigami for p=2, as a renormalized limit of modified p-energies on a sequence of graphs. Our proof is non-constructive, and does not settle the question of uniqueness. Based on the p-energy we may define p-harmonic functions as p-energy minimizers subject to boundary conditions, but again uniqueness is only conjectural. We present some numerical data as a complement to our results. This work is intended to pave the way for an eventual theory of p-Laplacians on fractals.

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Authors and Affiliations

  1. Mathematics Department, University of Chicago, Chicago, IL, 60637, USA (e-mail

    P. Edward Herman

  2. Dipartimento di Matematica, Universita di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Roma, Italy (e-mail

    Roberto Peirone

  3. Mathematics Department, Cornell University, Malott Hall, Ithaca, NY, 14853, USA (e-mail

    Robert S. Strichartz

Authors
  1. P. Edward Herman
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  2. Roberto Peirone
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  3. Robert S. Strichartz
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Herman, P.E., Peirone, R. & Strichartz, R.S. p-Energy and p-Harmonic Functions on Sierpinski Gasket Type Fractals. Potential Analysis 20, 125–148 (2004). https://doi.org/10.1023/A:1026377524793

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  • DOI: https://doi.org/10.1023/A:1026377524793

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