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

 

The resolvent kernel for PCF self-similar fractals


Authors: Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan and Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 362 (2010), 4451-4479
MSC (2010): Primary 28A80, 35P99, 47A75; Secondary 39A12, 39A70, 47B39
Published electronically: March 17, 2010
MathSciNet review: 2608413
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Abstract: For the Laplacian $ \Delta$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions and also with Neumann boundary conditions. That is, we construct a symmetric function $ G^{(\lambda)}$ which solves $ (\lambda \mathbb{I} - \Delta)^{-1} f(x) = \int G^{(\lambda)}(x,y) f(y) d\mu(y)$. The method is similar to Kigami's construction of the Green kernel and $ G^{(\lambda)}$ is expressed as a sum of scaled and ``translated'' copies of a certain function $ \psi^{(\lambda)}$ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $ SG_3$.


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Additional Information

Marius Ionescu
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14850-4201
Address at time of publication: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: mionescu@math.cornell.edu, ionescu@math.unconn.edu

Erin P. J. Pearse
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52246-1419
Email: erin-pearse@uiowa.edu

Luke G. Rogers
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: rogers@math.uconn.edu

Huo-Jun Ruan
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China – and – Department of Mathematics, Cornell University, Ithaca, New York 14850-4201
Email: ruanhj@zju.edu.cn

Robert S. Strichartz
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14850-4201
Email: str@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05098-1
PII: S 0002-9947(10)05098-1
Keywords: Dirichlet form, graph energy, discrete potential theory, discrete Laplace operator, graph Laplacian, eigenvalue, resolvent formula, post-critically finite, self-similar, fractal.
Received by editor(s): November 25, 2008
Received by editor(s) in revised form: April 20, 2009
Published electronically: March 17, 2010
Additional Notes: The work of the second author was partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.
The work of the fourth author was partially supported by grant NSFC 10601049 and by the Future Academic Star project of Zhejiang University.
The work of the fifth author was partially supported by NSF grant DMS 0652440.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.