Calculus on the Sierpinski gasket II: Point singularities, eigenfunctions, and normal derivatives of the heat kernel

Authors:
Nitsan Ben-Gal, Abby Shaw-Krauss, Robert S. Strichartz and Clint Young

Journal:
Trans. Amer. Math. Soc. **358** (2006), 3883-3936

MSC (2000):
Primary 28A80

DOI:
https://doi.org/10.1090/S0002-9947-06-04056-6

Published electronically:
April 17, 2006

MathSciNet review:
2219003

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Abstract: This paper continues the study of fundamental properties of elementary functions on the Sierpinski gasket (SG) related to the Laplacian defined by Kigami: harmonic functions, multiharmonic functions, and eigenfunctions of the Laplacian. We describe the possible point singularities of such functions, and we use the description at certain periodic points to motivate the definition of local derivatives at these points. We study the global behavior of eigenfunctions on all generic infinite blow-ups of SG, and construct eigenfunctions that decay at infinity in certain directions. We study the asymptotic behavior of normal derivatives of Dirichlet eigenfunctions at boundary points, and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points.

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Additional Information

**Nitsan Ben-Gal**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Address at time of publication:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Email:
Nitsan_Ben-Gal@brown.edu

**Abby Shaw-Krauss**

Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095

Address at time of publication:
Department of Applied Physics & Applied Mathematics, Columbia University, New York, New York 10027

Email:
ams2134@columbia.edu

**Robert S. Strichartz**

Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201

Email:
str@math.cornell.edu

**Clint Young**

Affiliation:
Department of Mathematics, SUNY Binghamton, Binghamton, New York 13902-6000

Address at time of publication:
Department of Physics, SUNY, Stony Brook, New York 11794

Email:
cyoung@grad.physics.sunysb.edu

DOI:
https://doi.org/10.1090/S0002-9947-06-04056-6

Received by editor(s):
June 8, 2004

Published electronically:
April 17, 2006

Additional Notes:
The first, second and fourth authors’ research was supported by the National Science Foundation through the Research Experiences for Undergraduates (REU) Program at Cornell University.

The third author’s research was supported in part by the National Science Foundation, grant DMS-0140194

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.