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

Ben-Gal, Nitsan; Shaw-Krauss, Abby; Strichartz, Robert; Young, Clint

doi:10.1090/S0002-9947-06-04056-6

Calculus on the Sierpinski gasket II: Point singularities, eigenfunctions, and normal derivatives of the heat kernel
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by Nitsan Ben-Gal, Abby Shaw-Krauss, Robert S. Strichartz and Clint Young PDF

Trans. Amer. Math. Soc. 358 (2006), 3883-3936 Request permission

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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