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January  2016, 15(1): 287-297. doi: 10.3934/cpaa.2016.15.287

The "hot spots" conjecture on higher dimensional Sierpinski gaskets

Xiao-Hui Li 1, and  Huo-Jun Ruan 1,

1. 

School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China, China

Received  April 2015 Revised  September 2015 Published  December 2015

In this paper, using spectral decimation, we prove that the ``hot spots" conjecture holds on higher dimensional Sierpinski gaskets.
Keywords: ``hot spots'' conjecture, eigenfunction, higher dimensional Sierpinski gaskets, spectral decimation., eigenvalue, Neumann Laplacian.
Mathematics Subject Classification: Primary: 28A80, 47A75; Secondary: 39A70, 47B3.
Citation: Xiao-Hui Li, Huo-Jun Ruan. The "hot spots" conjecture on higher dimensional Sierpinski gaskets. Communications on Pure & Applied Analysis, 2016, 15 (1) : 287-297. doi: 10.3934/cpaa.2016.15.287
References:
[1]

R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains,, \emph{J. Amer. Math. Soc.}, 17 (2004), 243.  doi: 10.1090/S0894-0347-04-00453-9.  Google Scholar

[2]

N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev, Vibration modes of $3n$-gaskets and other fractals,, \emph{J. Phys. A}, 41 (2008).  doi: 10.1088/1751-8113/41/1/015101.  Google Scholar

[3]

R. Bañuelos and K. Burdzy, On the "hot spots'' conjecture of J. Rauch,, \emph{J. Funct. Anal.}, 164 (1999), 1.  doi: 10.1006/jfan.1999.3397.  Google Scholar

[4]

K. Burdzy, The hot spots problem in planar domains with one hole,, \emph{Duke Math J.}, 129 (2005), 481.  doi: 10.1215/S0012-7094-05-12932-5.  Google Scholar

[5]

K. Burdzy and W. Werner, A counterexample to the "hot spots'' conjecture,, \emph{Ann. Math.}, 149 (1999), 309.  doi: 10.2307/121027.  Google Scholar

[6]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets,, \emph{Illinois J. Math.}, 53 (2009), 915.   Google Scholar

[7]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket,, \emph{Potential Anal.}, 1 (1992), 1.  doi: 10.1007/BF00249784.  Google Scholar

[8]

M. Ionescu, E. P. J. Pearse, L. G. Rogers, H.-J. Ruan and R. S. Strichartz, The resolvent kernel for PCF self-similar fractals,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4451.  doi: 10.1090/S0002-9947-10-05098-1.  Google Scholar

[9]

D. Jerison and N. Nadirashvili, The "hot spots'' conjecture for domains with two axes of symmetry,, \emph{J. Amer. Math. Soc.}, 13 (2000), 741.  doi: 10.1090/S0894-0347-00-00346-5.  Google Scholar

[10]

J. Kigami, A harmonic calculus on the Sierpinski Spaces,, \emph{Japan J. Appl. Math.}, 6 (1989), 259.  doi: 10.1007/BF03167882.  Google Scholar

[11]

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

[12]

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

[13]

H.-J. Ruan, The "hot spots" conjecture for the Sierpinski gasket,, \emph{Nonlinear Anal.}, 75 (2012), 469.  doi: 10.1016/j.na.2011.08.048.  Google Scholar

[14]

H.-J. Ruan and Y.-W. Zheng, The "hot spots" conjecture on the level-3 Sierpinski gasket,, \emph{Nonlinear Anal.}, 81 (2013), 101.  doi: 10.1016/j.na.2012.10.014.  Google Scholar

[15]

T. Shima, On eigenvalue problems for the random walks on the Sierpinski pre-gaskets,, \emph{Japan J. Indust. Appl. Math.}, 8 (1991), 127.  doi: 10.1007/BF03167188.  Google Scholar

[16]

T. Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets,, \emph{Japan J. Indust. Appl. Math.}, 13 (1996), 1.  doi: 10.1007/BF03167295.  Google Scholar

[17]

R. S. Strichartz, Differential Equations on Fractals,, Princeton University Press, (2006).   Google Scholar

show all references

References:
[1]

R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains,, \emph{J. Amer. Math. Soc.}, 17 (2004), 243.  doi: 10.1090/S0894-0347-04-00453-9.  Google Scholar

[2]

N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev, Vibration modes of $3n$-gaskets and other fractals,, \emph{J. Phys. A}, 41 (2008).  doi: 10.1088/1751-8113/41/1/015101.  Google Scholar

[3]

R. Bañuelos and K. Burdzy, On the "hot spots'' conjecture of J. Rauch,, \emph{J. Funct. Anal.}, 164 (1999), 1.  doi: 10.1006/jfan.1999.3397.  Google Scholar

[4]

K. Burdzy, The hot spots problem in planar domains with one hole,, \emph{Duke Math J.}, 129 (2005), 481.  doi: 10.1215/S0012-7094-05-12932-5.  Google Scholar

[5]

K. Burdzy and W. Werner, A counterexample to the "hot spots'' conjecture,, \emph{Ann. Math.}, 149 (1999), 309.  doi: 10.2307/121027.  Google Scholar

[6]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets,, \emph{Illinois J. Math.}, 53 (2009), 915.   Google Scholar

[7]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket,, \emph{Potential Anal.}, 1 (1992), 1.  doi: 10.1007/BF00249784.  Google Scholar

[8]

M. Ionescu, E. P. J. Pearse, L. G. Rogers, H.-J. Ruan and R. S. Strichartz, The resolvent kernel for PCF self-similar fractals,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4451.  doi: 10.1090/S0002-9947-10-05098-1.  Google Scholar

[9]

D. Jerison and N. Nadirashvili, The "hot spots'' conjecture for domains with two axes of symmetry,, \emph{J. Amer. Math. Soc.}, 13 (2000), 741.  doi: 10.1090/S0894-0347-00-00346-5.  Google Scholar

[10]

J. Kigami, A harmonic calculus on the Sierpinski Spaces,, \emph{Japan J. Appl. Math.}, 6 (1989), 259.  doi: 10.1007/BF03167882.  Google Scholar

[11]

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

[12]

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

[13]

H.-J. Ruan, The "hot spots" conjecture for the Sierpinski gasket,, \emph{Nonlinear Anal.}, 75 (2012), 469.  doi: 10.1016/j.na.2011.08.048.  Google Scholar

[14]

H.-J. Ruan and Y.-W. Zheng, The "hot spots" conjecture on the level-3 Sierpinski gasket,, \emph{Nonlinear Anal.}, 81 (2013), 101.  doi: 10.1016/j.na.2012.10.014.  Google Scholar

[15]

T. Shima, On eigenvalue problems for the random walks on the Sierpinski pre-gaskets,, \emph{Japan J. Indust. Appl. Math.}, 8 (1991), 127.  doi: 10.1007/BF03167188.  Google Scholar

[16]

T. Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets,, \emph{Japan J. Indust. Appl. Math.}, 13 (1996), 1.  doi: 10.1007/BF03167295.  Google Scholar

[17]

R. S. Strichartz, Differential Equations on Fractals,, Princeton University Press, (2006).   Google Scholar

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