Analysis of the Laplacian and spectral operators on the Vicsek set

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January 2011, 10(1): 1-44. doi: 10.3934/cpaa.2011.10.1

Analysis of the Laplacian and spectral operators on the Vicsek set

Sarah Constantin ^1, , Robert S. Strichartz ^2, and Miles Wheeler ^3,

1.	Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000, USA Government
2.	Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States
3.	Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USA Government

Received January 2010 Revised April 2010 Published November 2010

Abstract
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We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)

Keywords: Vicsek set, Laplacians on fractals, Green's function, eigenvalue ratio gaps., eigenvalue clusters, wave propagators, Weyl ratio, spectral operators, heat kernels.

Mathematics Subject Classification: 28A8.

Citation: Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure and Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1

References:

[1]	Bryant Adams, S. Alex Smith, Robert S. Strichartz and Alexander Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 1-24.
[2]	Adam Allan, Michael Barany and Robert S. Strichartz, Spectral operators on the Sierpinski gasket I, Complex variables and elliptic operators, 54 (2009), 521-543.
[3]	Tyrus Berry, Steven Heilman and Robert S. Strichartz, Outer approximation of the spectrum of a fractal Laplacian, Experimental Mathematics, 18 (2009), 449-480, arXiv:0904.3757.
[4]	Brian Bockelman and Robert S. Strichartz, Partial differential equations on products of Sierpinski gaskets, Indiana Univ. Math. J., 56 (2007), 1361-1375. doi: doi:10.1512/iumj.2007.56.2981.
[5]	Kevin Coletta, Kealey Dias and Robert S. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs' phenomenon, Fractals, 12 (2004), 413-449. doi: doi:10.1142/S0218348X04002689.
[6]	Sarah Constantin, Robert S. Strichartz and Wheeler Miles, Spectral operators on vicsek sets, 2009, http://www.math.cornell.edu/ mhw33.
[7]	Kyallee Dalrymple, Robert S. Strichartz and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284. doi: doi:10.1007/BF01261610.
[8]	S. Drenning and Robert S. Strichartz, Spectral decimation on Hambly's homogeneous, hierarchical gaskets, Ill. J. Math., 53 (2009), 915-937.
[9]	Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485. doi: doi:10.1016/S0022-1236(02)00009-5.
[10]	Taryn Flock and Robsert S. Strichartz, Laplacians on a family of quadratic Julia sets, Trans. Amer. Math. Soc., to appear.
[11]	M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: doi:10.1007/BF00249784.
[12]	Peter J. Grabner and Wolfgang Woess, Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph, Stochastic Process. Appl., 69 (1997), 127-138. doi: doi:10.1016/S0304-4149(97)00033-1.
[13]	A. Grigor'yan and L. Saloff-Coste, Heat kernels on manifolds with ends, Ann. Inst. Fourier, 59 (2009), 1917-1997.
[14]	Kathryn E. Hare and Denglin Zhou, Gaps in the ratios of the spectrum of Laplacians on fractals, Fractals, 17 (2009), 523-535. doi: doi:10.1142/S0218348X0900451X.
[15]	Jun Kigami, "Analysis on Fractals," Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001.
[16]	Jun Kigami and Michel L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125. doi: doi:10.1007/BF02097233.
[17]	Richard Oberlin, Brian Street and Robert S. Strichartz, Sampling on the Sierpinski gasket, Experiment. Math., 12 (2003), 403-418.
[18]	Adam Sikora, Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., 58 (2009), 317-334. doi: doi:10.1512/iumj.2009.58.3745.
[19]	Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
[20]	Robert S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series, Math. Res. Lett., 12 (2005), 269-274.
[21]	Robert S. Strichartz, "Differential Equations on Fractals: A Tutorial," Princeton University Press, Princeton, NJ, 2006.
[22]	Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567. doi: doi:10.1006/jfan.1998.3297.
[23]	Denglin Zhou, Spectral analysis of Laplacians on the Vicsek set, Pacific J. Math., 241 (2009), 369-398. doi: doi:10.2140/pjm.2009.241.369.

show all references

References:

[1]	Bryant Adams, S. Alex Smith, Robert S. Strichartz and Alexander Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 1-24.
[2]	Adam Allan, Michael Barany and Robert S. Strichartz, Spectral operators on the Sierpinski gasket I, Complex variables and elliptic operators, 54 (2009), 521-543.
[3]	Tyrus Berry, Steven Heilman and Robert S. Strichartz, Outer approximation of the spectrum of a fractal Laplacian, Experimental Mathematics, 18 (2009), 449-480, arXiv:0904.3757.
[4]	Brian Bockelman and Robert S. Strichartz, Partial differential equations on products of Sierpinski gaskets, Indiana Univ. Math. J., 56 (2007), 1361-1375. doi: doi:10.1512/iumj.2007.56.2981.
[5]	Kevin Coletta, Kealey Dias and Robert S. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs' phenomenon, Fractals, 12 (2004), 413-449. doi: doi:10.1142/S0218348X04002689.
[6]	Sarah Constantin, Robert S. Strichartz and Wheeler Miles, Spectral operators on vicsek sets, 2009, http://www.math.cornell.edu/ mhw33.
[7]	Kyallee Dalrymple, Robert S. Strichartz and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284. doi: doi:10.1007/BF01261610.
[8]	S. Drenning and Robert S. Strichartz, Spectral decimation on Hambly's homogeneous, hierarchical gaskets, Ill. J. Math., 53 (2009), 915-937.
[9]	Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485. doi: doi:10.1016/S0022-1236(02)00009-5.
[10]	Taryn Flock and Robsert S. Strichartz, Laplacians on a family of quadratic Julia sets, Trans. Amer. Math. Soc., to appear.
[11]	M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: doi:10.1007/BF00249784.
[12]	Peter J. Grabner and Wolfgang Woess, Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph, Stochastic Process. Appl., 69 (1997), 127-138. doi: doi:10.1016/S0304-4149(97)00033-1.
[13]	A. Grigor'yan and L. Saloff-Coste, Heat kernels on manifolds with ends, Ann. Inst. Fourier, 59 (2009), 1917-1997.
[14]	Kathryn E. Hare and Denglin Zhou, Gaps in the ratios of the spectrum of Laplacians on fractals, Fractals, 17 (2009), 523-535. doi: doi:10.1142/S0218348X0900451X.
[15]	Jun Kigami, "Analysis on Fractals," Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001.
[16]	Jun Kigami and Michel L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125. doi: doi:10.1007/BF02097233.
[17]	Richard Oberlin, Brian Street and Robert S. Strichartz, Sampling on the Sierpinski gasket, Experiment. Math., 12 (2003), 403-418.
[18]	Adam Sikora, Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., 58 (2009), 317-334. doi: doi:10.1512/iumj.2009.58.3745.
[19]	Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
[20]	Robert S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series, Math. Res. Lett., 12 (2005), 269-274.
[21]	Robert S. Strichartz, "Differential Equations on Fractals: A Tutorial," Princeton University Press, Princeton, NJ, 2006.
[22]	Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567. doi: doi:10.1006/jfan.1998.3297.
[23]	Denglin Zhou, Spectral analysis of Laplacians on the Vicsek set, Pacific J. Math., 241 (2009), 369-398. doi: doi:10.2140/pjm.2009.241.369.

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