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Restrictions of Laplacian Eigenfunctions to Edges in the Sierpinski Gasket

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Abstract

In this paper, we study the restrictions of both the harmonic functions and the eigenfunctions of the symmetric Laplacian to edges of pre-gaskets contained in the Sierpinski gasket. For a harmonic function, its restriction to any edge is either monotone or having a single extremum. For an eigenfunction, it may have several local extrema along edges. We prove general criteria, involving the values of any given function at the endpoints and midpoint of any edge, to determine which case it should be, as well as the asymptotic behavior of the restriction near the endpoints. Moreover, for eigenfunctions, we use spectral decimation to calculate the exact numbers of the local extrema along edges. This confirms, in a more general situation, a conjecture of Dalrymple et al. (J Fourier Anal Appl 5:203–284, 1999) on the behavior of the restrictions to edges of the basis Dirichlet eigenfunctions, suggested by the numerical data.

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Acknowledgements

We are grateful to the anonymous referee for several important suggestions that led to the improvement of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu, People’s Republic of China

    Hua Qiu & Haoran Tian

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  1. Hua Qiu
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  2. Haoran Tian
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Correspondence to Haoran Tian.

Additional information

Communicated by Jeff Geronimo.

The research of the first author was supported by the National Natural Science Foundation of China, Grant 11471157.

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Qiu, H., Tian, H. Restrictions of Laplacian Eigenfunctions to Edges in the Sierpinski Gasket. Constr Approx 50, 243–269 (2019). https://doi.org/10.1007/s00365-018-9451-5

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  • DOI: https://doi.org/10.1007/s00365-018-9451-5

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