July  2020, 19(7): 3901-3916. doi: 10.3934/cpaa.2020159

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

Received  July 2019 Revised  January 2020 Published  April 2020

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.

Citation: Shiping Cao, Shuangping Li, Robert S. Strichartz, Prem Talwai. A trace theorem for Sobolev spaces on the Sierpinski gasket. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3901-3916. doi: 10.3934/cpaa.2020159
References:
[1]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Field, 79 (1988), 543-623.  doi: 10.1007/BF00318785.

[2]

S. Cao and H. Qiu, Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions, in submission.

[3]

S. Cao and H. Qiu, Boundary Value Problems for harmonic functions on domains in Sierpinski gaskets, Commun. Pure Appl. Anal., 19 (2020), 1147-1179.  doi: 10.3934/cpaa.2020054.

[4]

S. Cao and H. Qiu, Higher order tangents and Higher order Laplacians on Sierpinski Gasket Type Fractals, preprint, arXiv: 1607.07544.

[5]

Q. Gu and K. Lau, Dirichlet forms and critical exponents on fractals, preprint, arXiv: 1703.07061. doi: 10.1090/tran/8004.

[6]

Z. GuoR. KoganH. Qiu and R. S. Strichartz, Boundary value problems for a family of domains in the Sierpinski gasket, Illinois J. Math., 58 (2014), 497-519. 

[7]

M. Hino and T. Kumagai, A trace theorem for Dirichlet forms on fractals, J. Func. Anal., 238 (2006), 578-611.  doi: 10.1016/j.jfa.2006.05.012.

[8]

M. Hinz, D. Koch and M. Meinert, Sobolev spaces and calculus of variations on fractals, preprint, arXiv: 1805.04456.

[9]

J. Hu and X. Wang, Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals, Studia Math., 177 (2006), 153-172.  doi: 10.4064/sm177-2-5.

[10]

J. Hu and M. Zähle, Potential spaces on fractals, Studia Math., 170 (2005), 259-281.  doi: 10.4064/sm170-3-4.

[11]

M. IonescuL. G. Rogers and R. S. Strichartz, Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.  doi: 10.4171/RMI/752.

[12]

A. Jonsson, A trace theorem for the Dirichlet form on the Sierpinski gasket, Math. Z., 250 (2005), 599-609.  doi: 10.1007/s00209-005-0767-z.

[13]

A. Jonsson, Brownian motion on fractals and function spaces, Math. Z., 222 (1996), 495-504.  doi: 10.1007/PL00004543.

[14]

A. Kamont, A discrete characterization of Besov Spaces, Approx. Theory Appl., 13 (1997), 63-77. 

[15]

J. Kigami, A harmonic calculus on the Sierpinski spaces, Jpn. J. Appl. Math., 6 (1989), 259-290.  doi: 10.1007/BF03167882.

[16]

J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.  doi: 10.2307/2154402.

[17]

J. Kigami, Analysis on Fractals, Cambridge University Press, 2001. doi: 10.1017/CBO9780511470943.

[18]

T. Kumagai, Brownian Motion Penetrating Fractals: An Application of the Trace Theorem of Besov Spaces, J. Func. Anal., 170 (2000), 69-92.  doi: 10.1006/jfan.1999.3500.

[19]

W. Li and R. S. Strichartz, Boundary value problems on a half Sierpinski gasket, J. Fractal Geom., 1 (2014), 1-43.  doi: 10.4171/JFG/1.

[20]

T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990). doi: 10.1090/memo/0420.

[21]

J. Owen and R. S. Strichartz, Boundary value problems for harmonic functions on a domain in the Sierpinski gasket, Indiana Univ. Math. J., 61 (2012), 319-335.  doi: 10.1512/iumj.2012.61.4539.

[22]

H. Qiu, Exact spectrum of the Laplacian on a domain in the Sierpinski gasket, J. Funct. Anal., 277 (2019), 806-888.  doi: 10.1016/j.jfa.2018.08.018.

[23]

R. S. Strichartz and C. Wong, The p-Laplacian on the Sierpinski gasket, Nonlinearity, 17 (2004), 595-616.  doi: 10.1088/0951-7715/17/2/014.

[24] R. S. Strichartz, Differential Equations on Fractals. A Tutorial, Princeton University Press, 2006. 
[25]

R. S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83.  doi: 10.1016/S0022-1236(02)00035-6.

show all references

References:
[1]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Field, 79 (1988), 543-623.  doi: 10.1007/BF00318785.

[2]

S. Cao and H. Qiu, Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions, in submission.

[3]

S. Cao and H. Qiu, Boundary Value Problems for harmonic functions on domains in Sierpinski gaskets, Commun. Pure Appl. Anal., 19 (2020), 1147-1179.  doi: 10.3934/cpaa.2020054.

[4]

S. Cao and H. Qiu, Higher order tangents and Higher order Laplacians on Sierpinski Gasket Type Fractals, preprint, arXiv: 1607.07544.

[5]

Q. Gu and K. Lau, Dirichlet forms and critical exponents on fractals, preprint, arXiv: 1703.07061. doi: 10.1090/tran/8004.

[6]

Z. GuoR. KoganH. Qiu and R. S. Strichartz, Boundary value problems for a family of domains in the Sierpinski gasket, Illinois J. Math., 58 (2014), 497-519. 

[7]

M. Hino and T. Kumagai, A trace theorem for Dirichlet forms on fractals, J. Func. Anal., 238 (2006), 578-611.  doi: 10.1016/j.jfa.2006.05.012.

[8]

M. Hinz, D. Koch and M. Meinert, Sobolev spaces and calculus of variations on fractals, preprint, arXiv: 1805.04456.

[9]

J. Hu and X. Wang, Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals, Studia Math., 177 (2006), 153-172.  doi: 10.4064/sm177-2-5.

[10]

J. Hu and M. Zähle, Potential spaces on fractals, Studia Math., 170 (2005), 259-281.  doi: 10.4064/sm170-3-4.

[11]

M. IonescuL. G. Rogers and R. S. Strichartz, Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.  doi: 10.4171/RMI/752.

[12]

A. Jonsson, A trace theorem for the Dirichlet form on the Sierpinski gasket, Math. Z., 250 (2005), 599-609.  doi: 10.1007/s00209-005-0767-z.

[13]

A. Jonsson, Brownian motion on fractals and function spaces, Math. Z., 222 (1996), 495-504.  doi: 10.1007/PL00004543.

[14]

A. Kamont, A discrete characterization of Besov Spaces, Approx. Theory Appl., 13 (1997), 63-77. 

[15]

J. Kigami, A harmonic calculus on the Sierpinski spaces, Jpn. J. Appl. Math., 6 (1989), 259-290.  doi: 10.1007/BF03167882.

[16]

J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.  doi: 10.2307/2154402.

[17]

J. Kigami, Analysis on Fractals, Cambridge University Press, 2001. doi: 10.1017/CBO9780511470943.

[18]

T. Kumagai, Brownian Motion Penetrating Fractals: An Application of the Trace Theorem of Besov Spaces, J. Func. Anal., 170 (2000), 69-92.  doi: 10.1006/jfan.1999.3500.

[19]

W. Li and R. S. Strichartz, Boundary value problems on a half Sierpinski gasket, J. Fractal Geom., 1 (2014), 1-43.  doi: 10.4171/JFG/1.

[20]

T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990). doi: 10.1090/memo/0420.

[21]

J. Owen and R. S. Strichartz, Boundary value problems for harmonic functions on a domain in the Sierpinski gasket, Indiana Univ. Math. J., 61 (2012), 319-335.  doi: 10.1512/iumj.2012.61.4539.

[22]

H. Qiu, Exact spectrum of the Laplacian on a domain in the Sierpinski gasket, J. Funct. Anal., 277 (2019), 806-888.  doi: 10.1016/j.jfa.2018.08.018.

[23]

R. S. Strichartz and C. Wong, The p-Laplacian on the Sierpinski gasket, Nonlinearity, 17 (2004), 595-616.  doi: 10.1088/0951-7715/17/2/014.

[24] R. S. Strichartz, Differential Equations on Fractals. A Tutorial, Princeton University Press, 2006. 
[25]

R. S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83.  doi: 10.1016/S0022-1236(02)00035-6.

Figure 1.  the Sierpinski gasket
Figure 2.  The harmonic function with $ h(q_0) = a, h(q_1) = b, h(q_2) = c $
Figure 3.  The points $ x_{(n, k)} = F_{w(n, k)}q_0 $
Figure 4.  An illustration for $ Z_{n, k} $ and $ \tilde{Z}_{n, k} $
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