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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

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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