The resolvent kernel for PCF selfsimilar fractals
Authors:
Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, HuoJun Ruan and Robert S. Strichartz
Journal:
Trans. Amer. Math. Soc. 362 (2010), 44514479
MSC (2010):
Primary 28A80, 35P99, 47A75; Secondary 39A12, 39A70, 47B39
Published electronically:
March 17, 2010
MathSciNet review:
2608413
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Similar Articles 
Additional Information
Abstract: For the Laplacian defined on a p.c.f. selfsimilar 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 which solves . The method is similar to Kigami's construction of the Green kernel and is expressed as a sum of scaled and ``translated'' copies of a certain function 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 level3 Sierpinski gasket .
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Khalil, P.
Mody, B.
Steinhurst, and A.
Teplyaev, Vibration modes of 3𝑛gaskets and other
fractals, J. Phys. A 41 (2008), no. 1, 015101,
21. MR
2450694 (2010a:28008), 10.1088/17518113/41/1/015101
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Jessica
L. DeGrado, Luke
G. Rogers, and Robert
S. Strichartz, Gradients of Laplacian eigenfunctions
on the Sierpinski gasket, Proc. Amer. Math.
Soc. 137 (2009), no. 2, 531–540. MR 2448573
(2009k:28018), 10.1090/S0002993908097116
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Sean Drenning and Robert S. Strichartz.
Spectral decimation on Hambly's homogeneous hierarchical gaskets. To appear in: Illinois J. Math.
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J. Fitzsimmons, Ben
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Kumagai, Transition density estimates for Brownian motion on affine
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(95j:60122)
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Luke Rogers.
Estimates for the resolvent kernel for PCF selfsimilar fractals. In preparation, 2008.
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Sabot, Existence and uniqueness of diffusions on finitely ramified
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Denglin Zhou.
Criteria for spectral gaps of Laplacians on fractals. J. Fourier Anal. Appl., 16(1):7697, 2010.
 [BCD08]
 Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, Michael Khalil, Poorak Mody, Benjamin Steinhurst, and Alexander Teplyaev.
Vibration modes of 3gaskets and other fractals. J. Phys. A: Math. Theor., 41:015101 (21pp), 2008. MR 2450694
 [DRS09]
 Jessica L. DeGrado, Luke G. Rogers, and Robert S. Strichartz.
Gradients of Laplacian eigenfunctions on the Sierpinski gasket. Proc. Amer. Math. Soc., (137):531540, 2009. MR 2448573
 [DS07]
 Sean Drenning and Robert S. Strichartz.
Spectral decimation on Hambly's homogeneous hierarchical gaskets. To appear in: Illinois J. Math.
 [FHK94]
 Pat J. Fitzsimmons, Ben M. Hambly, and Takashi Kumagai.
Transition density estimates for Brownian motion on affine nested fractals. Comm. Math. Phys., 165(3):595620, 1994. MR 1301625 (95j:60122)
 [HK99]
 Ben M. Hambly and Takashi Kumagai.
Transition density estimates for diffusion processes on post critically finite selfsimilar fractals. Proc. London Math. Soc. (3), 78(2):431458, 1999. MR 1665249 (99m:60118)
 [Hut81]
 John E. Hutchinson.
Fractals and selfsimilarity. Indiana Univ. Math. J., 30(5):713747, 1981. MR 625600 (82h:49026)
 [Kig01]
 Jun Kigami.
Analysis on fractals, volume 143 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
 [Kig03]
 Jun Kigami.
Harmonic analysis for resistance forms. J. Funct. Anal., 204(2):399444, 2003. MR 2017320 (2004m:31010)
 [Rog08]
 Luke Rogers.
Estimates for the resolvent kernel for PCF selfsimilar fractals. In preparation, 2008.
 [Sab97]
 Christophe Sabot.
Existence and uniqueness of diffusions on finitely ramified selfsimilar fractals. Ann. Sci. École Norm. Sup. (4), 30(5):605673, 1997. MR 1474807 (98h:60118)
 [See67]
 Robert T. Seeley.
Complex powers of an elliptic operator. In Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pages 288307. Amer. Math. Soc., Providence, R.I., 1967. MR 0237943 (38:6220)
 [See69]
 Robert T. Seeley.
The resolvent of an elliptic boundary problem. Amer. J. Math., 91:889920, 1969. MR 0265764 (42:673)
 [Str06]
 Robert S. Strichartz.
Differential equations on fractals. Princeton University Press, Princeton, NJ, 2006. A tutorial. MR 2246975 (2007f:35003)
 [Tep98]
 Alexander Teplyaev.
Spectral analysis on infinite Sierpiński gaskets. J. Funct. Anal., 159(2):537567, 1998. MR 1658094 (99j:35153)
 [Zho09]
 Denglin Zhou.
Spectral analysis of Laplacians on the Vicsek set. Pacific J. Math., 241(2):369398, 2009. MR 2507583
 [Zho10]
 Denglin Zhou.
Criteria for spectral gaps of Laplacians on fractals. J. Fourier Anal. Appl., 16(1):7697, 2010.
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Additional Information
Marius Ionescu
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 148504201
Address at time of publication:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 062693009
Email:
mionescu@math.cornell.edu, ionescu@math.unconn.edu
Erin P. J. Pearse
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 522461419
Email:
erinpearse@uiowa.edu
Luke G. Rogers
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 062693009
Email:
rogers@math.uconn.edu
HuoJun Ruan
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China – and – Department of Mathematics, Cornell University, Ithaca, New York 148504201
Email:
ruanhj@zju.edu.cn
Robert S. Strichartz
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 148504201
Email:
str@math.cornell.edu
DOI:
http://dx.doi.org/10.1090/S0002994710050981
Keywords:
Dirichlet form,
graph energy,
discrete potential theory,
discrete Laplace operator,
graph Laplacian,
eigenvalue,
resolvent formula,
postcritically finite,
selfsimilar,
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 DMS0602242.
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.
