A trace theorem for Sobolev spaces on the Sierpinski gasket

Shiping Cao; Shuangping Li; Robert S. Strichartz; Prem Talwai

doi:10.3934/cpaa.2020159

Article Contents

2020, Volume 19, Issue 7: 3901-3916. Doi: 10.3934/cpaa.2020159

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A trace theorem for Sobolev spaces on the Sierpinski gasket

1.
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
2.
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA
3.
Decision, Risk, and Operations, Columbia Business School, New York, NY 10027, USA

^* Corresponding author
^* Corresponding author

Received: June 30, 2019

Revised: December 31, 2019

Published: April 2020

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Abstract

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the $ L^2 $ domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.

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Mathematics Subject Classification: Primary 28A80.

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Figure 1. the Sierpinski gasket

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Figure 2. The harmonic function with $ h(q_0) = a, h(q_1) = b, h(q_2) = c $

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Figure 3. The points $ x_{(n, k)} = F_{w(n, k)}q_0 $

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Figure 4. An illustration for $ Z_{n, k} $ and $ \tilde{Z}_{n, k} $

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