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

Azzam, Jonas; Hall, Michael; Strichartz, Robert

doi:10.1090/S0002-9947-07-04363-2

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket
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by Jonas Azzam, Michael A. Hall and Robert S. Strichartz PDF

Trans. Amer. Math. Soc. 360 (2008), 2089-2130 Request permission

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 $n\neq 2$. 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 $p\geq 2$. We derive analogous linear extension algorithms for energy measures on related fractals.

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