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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For the Laplacian 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 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 level-3 Sierpinski gasket .

**[BCD08]**N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst, and A. Teplyaev,*Vibration modes of 3𝑛-gaskets and other fractals*, J. Phys. A**41**(2008), no. 1, 015101, 21. MR**2450694**, 10.1088/1751-8113/41/1/015101**[DRS09]**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**, 10.1090/S0002-9939-08-09711-6**[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**(1994), no. 3, 595–620. MR**1301625****[HK99]**B. M. Hambly and T. Kumagai,*Transition density estimates for diffusion processes on post critically finite self-similar fractals*, Proc. London Math. Soc. (3)**78**(1999), no. 2, 431–458. MR**1665249**, 10.1112/S0024611599001744**[Hut81]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[Kig01]**Jun Kigami,*Analysis on fractals*, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR**1840042****[Kig03]**Jun Kigami,*Harmonic analysis for resistance forms*, J. Funct. Anal.**204**(2003), no. 2, 399–444. MR**2017320**, 10.1016/S0022-1236(02)00149-0**[Rog08]**Luke Rogers.

Estimates for the resolvent kernel for PCF self-similar fractals.*In preparation*, 2008.**[Sab97]**C. Sabot,*Existence and uniqueness of diffusions on finitely ramified self-similar fractals*, Ann. Sci. École Norm. Sup. (4)**30**(1997), no. 5, 605–673 (English, with English and French summaries). MR**1474807**, 10.1016/S0012-9593(97)89934-X**[See67]**R. T. Seeley,*Complex powers of an elliptic operator*, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR**0237943****[See69]**R. Seeley,*The resolvent of an elliptic boundary problem*, Amer. J. Math.**91**(1969), 889–920. MR**0265764****[Str06]**Robert S. Strichartz,*Differential equations on fractals*, Princeton University Press, Princeton, NJ, 2006. A tutorial. MR**2246975****[Tep98]**Alexander Teplyaev,*Spectral analysis on infinite Sierpiński gaskets*, J. Funct. Anal.**159**(1998), no. 2, 537–567. MR**1658094**, 10.1006/jfan.1998.3297**[Zho09]**Denglin Zhou,*Spectral analysis of Laplacians on the Vicsek set*, Pacific J. Math.**241**(2009), no. 2, 369–398. MR**2507583**, 10.2140/pjm.2009.241.369**[Zho10]**Denglin Zhou.

Criteria for spectral gaps of Laplacians on fractals.*J. Fourier Anal. Appl.*, 16(1):76-97, 2010.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
28A80,
35P99,
47A75,
39A12,
39A70,
47B39

Retrieve articles in all journals with MSC (2010): 28A80, 35P99, 47A75, 39A12, 39A70, 47B39

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:
https://doi.org/10.1090/S0002-9947-10-05098-1

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.