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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket


Authors: Jonas Azzam, Michael A. Hall and Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 360 (2008), 2089-2130
MSC (2000): Primary 28A80
Published electronically: November 28, 2007
MathSciNet review: 2366976
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Abstract: On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy $ \mathcal{E}_\varphi$ and conformal Laplacian $ \Delta_{\varphi}$ for a given conformal factor $ \varphi$, based on the corresponding notions in Riemannian geometry in dimension $ n\neq2$. We derive a differential equation that describes the dependence of the effective resistances of $ \mathcal{E}_\varphi$ on $ \varphi$. We show that the spectrum of $ \Delta_{\varphi}$ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function $ \varphi$ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the $ L^{p}$ dimensions of these measures for integer values of $ p\geq2$. We derive analogous linear extension algorithms for energy measures on related fractals.


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Additional Information

Jonas Azzam
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Address at time of publication: Department of Mathematics, UCLA, Los Angeles, California 90095
Email: jonasazzam@ucla.edu

Michael A. Hall
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Department of Mathematics, UCLA, Los Angeles, California 90095
Email: michaelhall@ucla.edu

Robert S. Strichartz
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
Email: str@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04363-2
PII: S 0002-9947(07)04363-2
Received by editor(s): March 27, 2006
Published electronically: November 28, 2007
Additional Notes: The first and second authors were supported by the National Science Foundation through the Research Experiences for Undergraduates (REU) program at Cornell.
The third author was supported in part by the National Science Foundation, grant DMS-0140194.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.