The resolvent kernel for PCF self-similar fractals
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- by Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan and Robert S. Strichartz PDF
- Trans. Amer. Math. Soc. 362 (2010), 4451-4479 Request permission
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 $SG_3$.References
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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
- MR Author ID: 785199
- 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
- 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. - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4451-4479
- MSC (2010): Primary 28A80, 35P99, 47A75; Secondary 39A12, 39A70, 47B39
- DOI: https://doi.org/10.1090/S0002-9947-10-05098-1
- MathSciNet review: 2608413