American Institute of Mathematical Sciences

Advanced
February  2020, 19(2): 1147-1179. doi: 10.3934/cpaa.2020054

Boundary value problems for harmonic functions on domains in Sierpinski gaskets

Shiping Cao 1, and  Hua Qiu 2,,

1. 

Department of Mathematics, Cornell Univerisity, Ithaca 14853, USA
2. 

Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China

* Corresponding author

Received  May 2019 Revised  July 2019 Published  October 2019

Fund Project: The research of the second author was supported by Nature Science Foundation of China, Grant 11471157.

We study boundary value problems for harmonic functions on certain domains in the level-$ l $ Sierpinski gaskets $ \mathcal{SG}_l $($ l\geq 2 $) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of $ \mathcal{SG}_l $ and the upper and lower domains generated by horizontal cuts of $ \mathcal{SG}_l $ are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work.

Keywords: Boundary value problems, harmonic functions, fractal Laplacians, Sierpinski gasket, energy estimates.
Mathematics Subject Classification: Primary: 28A80; Secondary: 35J25.
Citation: Shiping Cao, Hua Qiu. Boundary value problems for harmonic functions on domains in Sierpinski gaskets. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1147-1179. doi: 10.3934/cpaa.2020054
References:
[1]

Z. Guo, R. Kogan, H. 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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

J. Kigami, Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees, Adv. Math., 225 (2010), 2674-2730.  doi: 10.1016/j.aim.2010.04.029.  Google Scholar

[8]

J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012). doi: 10.1090/S0065-9266-2011-00632-5.  Google Scholar

[9]

P. H. Li, N. Ryder, R. S. Strichartz and B. E. Ugurcan, Extensions and their minimizations on the Sierpinski gasket, Potential Anal., 41 (2014), 1167-1201.  doi: 10.1007/s11118-014-9415-8.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

R. S. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal., 164 (1999), 181-208.  doi: 10.1006/jfan.1999.3400.  Google Scholar

[14]

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

[15]

R. S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton University Press, 2006.  Google Scholar

[16]

R.S. Strichartz, "Graph paper" trace characterizations of functions of finite energy, J. Anal. Math., 128 (2016), 239-260.  doi: 10.1007/s11854-016-0008-x.  Google Scholar

show all references

References:
[1]

Z. Guo, R. Kogan, H. 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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

J. Kigami, Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees, Adv. Math., 225 (2010), 2674-2730.  doi: 10.1016/j.aim.2010.04.029.  Google Scholar

[8]

J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012). doi: 10.1090/S0065-9266-2011-00632-5.  Google Scholar

[9]

P. H. Li, N. Ryder, R. S. Strichartz and B. E. Ugurcan, Extensions and their minimizations on the Sierpinski gasket, Potential Anal., 41 (2014), 1167-1201.  doi: 10.1007/s11118-014-9415-8.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

R. S. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal., 164 (1999), 181-208.  doi: 10.1006/jfan.1999.3400.  Google Scholar

[14]

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

[15]

R. S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton University Press, 2006.  Google Scholar

[16]

R.S. Strichartz, "Graph paper" trace characterizations of functions of finite energy, J. Anal. Math., 128 (2016), 239-260.  doi: 10.1007/s11854-016-0008-x.  Google Scholar

Figure 1.  Upper and lower domains in $ \mathcal{SG} $ and $ \mathcal{SG}_3 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Half domains in $ \mathcal{SG} $ and $ \mathcal{SG}_3 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  $ \Gamma_1,\Gamma_2,\Gamma_3 $ of $ \mathcal{SG}_3 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Harmonic extension algorithm of $ \mathcal{SG}_3 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The values of $ h_a $ on $ V_1\cap\bar{\Omega} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Simple sets $ O_1,O_2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The values of $ v $ on $ O_2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The upper domain
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  The relationship between $ \Omega_{\lambda_0} $ and $ \Omega_{\lambda_1} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  The upper domain $ \Omega_{0.39} $. The shaded regions are $ F^\lambda_5\Omega_{\lambda_1} $, $ F^\lambda_{54}\Omega_{\lambda_2} $, $ F^\lambda_{543}\Omega_{\lambda_3} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Boundary values of $ h^{(1)} $, $ h^{(2)} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  Two typical domain $ \Omega_\lambda^- $'s. $ \lambda $ is (or not) a dyadic rational
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  The relationship between $ \Omega_\lambda^- $ and $ \Omega_{S\lambda}^- $. $ e_1(\lambda) = 0 $ in the left one and $ e_1(\lambda) = 1 $ in the right one
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 14.  $ h_1+h_2 $ and $ h_1-h_2 $ in two cases
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 15.  $ V_m^\lambda $ and some conductances. ($ \frac{5}{8}<\lambda<\frac{3}{4},m = 3 $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 16.  The half domain of $ \mathcal{SG}_4 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 17.  The vertices on a half $ \mathcal{SG} $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke Rogers, Luis Seda. Magnetic Laplacians of locally exact forms on the Sierpinski Gasket. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2299-2319. doi: 10.3934/cpaa.2017113
[2]

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
[3]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43
[4]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
[5]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779
[6]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425
[7]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
[8]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[9]

Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51
[10]

Xi-Nan Ma, Jiang Ye, Yun-Hua Ye. Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 225-243. doi: 10.3934/cpaa.2011.10.225
[11]

Laura Abatangelo, Susanna Terracini. Harmonic functions in union of chambers. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5609-5629. doi: 10.3934/dcds.2015.35.5609
[12]

Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403
[13]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[14]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[15]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[16]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377
[17]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283
[18]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381
[19]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[20]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (193)
  • HTML views (81)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy