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

The "hot spots" conjecture on higher dimensional Sierpinski gaskets

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.
Citation: Xiao-Hui Li, Huo-Jun Ruan. The "hot spots" conjecture on higher dimensional Sierpinski gaskets. Communications on Pure and 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, J. Amer. Math. Soc., 17 (2004), 243-265. doi: 10.1090/S0894-0347-04-00453-9.

[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, J. Phys. A, 41 (2008), 015101. doi: 10.1088/1751-8113/41/1/015101.

[3]

R. Bañuelos and K. Burdzy, On the "hot spots'' conjecture of J. Rauch, J. Funct. Anal., 164 (1999), 1-33. doi: 10.1006/jfan.1999.3397.

[4]

K. Burdzy, The hot spots problem in planar domains with one hole, Duke Math J., 129 (2005), 481-502. doi: 10.1215/S0012-7094-05-12932-5.

[5]

K. Burdzy and W. Werner, A counterexample to the "hot spots'' conjecture, Ann. Math., 149 (1999), 309-317. doi: 10.2307/121027.

[6]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets, Illinois J. Math., 53 (2009), 915-937.

[7]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784.

[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, Trans. Amer. Math. Soc., 362 (2010), 4451-4479. doi: 10.1090/S0002-9947-10-05098-1.

[9]

D. Jerison and N. Nadirashvili, The "hot spots'' conjecture for domains with two axes of symmetry, J. Amer. Math. Soc., 13 (2000), 741-772. doi: 10.1090/S0894-0347-00-00346-5.

[10]

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

[11]

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

[12]

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

[13]

H.-J. Ruan, The "hot spots" conjecture for the Sierpinski gasket, Nonlinear Anal., 75 (2012), 469-476. doi: 10.1016/j.na.2011.08.048.

[14]

H.-J. Ruan and Y.-W. Zheng, The "hot spots" conjecture on the level-3 Sierpinski gasket, Nonlinear Anal., 81 (2013), 101-109. doi: 10.1016/j.na.2012.10.014.

[15]

T. Shima, On eigenvalue problems for the random walks on the Sierpinski pre-gaskets, Japan J. Indust. Appl. Math., 8 (1991), 127-141. doi: 10.1007/BF03167188.

[16]

T. Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math., 13 (1996), 1-23. doi: 10.1007/BF03167295.

[17]

R. S. Strichartz, Differential Equations on Fractals, Princeton University Press, 2006.

show all references

References:
[1]

R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc., 17 (2004), 243-265. doi: 10.1090/S0894-0347-04-00453-9.

[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, J. Phys. A, 41 (2008), 015101. doi: 10.1088/1751-8113/41/1/015101.

[3]

R. Bañuelos and K. Burdzy, On the "hot spots'' conjecture of J. Rauch, J. Funct. Anal., 164 (1999), 1-33. doi: 10.1006/jfan.1999.3397.

[4]

K. Burdzy, The hot spots problem in planar domains with one hole, Duke Math J., 129 (2005), 481-502. doi: 10.1215/S0012-7094-05-12932-5.

[5]

K. Burdzy and W. Werner, A counterexample to the "hot spots'' conjecture, Ann. Math., 149 (1999), 309-317. doi: 10.2307/121027.

[6]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets, Illinois J. Math., 53 (2009), 915-937.

[7]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784.

[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, Trans. Amer. Math. Soc., 362 (2010), 4451-4479. doi: 10.1090/S0002-9947-10-05098-1.

[9]

D. Jerison and N. Nadirashvili, The "hot spots'' conjecture for domains with two axes of symmetry, J. Amer. Math. Soc., 13 (2000), 741-772. doi: 10.1090/S0894-0347-00-00346-5.

[10]

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

[11]

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

[12]

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

[13]

H.-J. Ruan, The "hot spots" conjecture for the Sierpinski gasket, Nonlinear Anal., 75 (2012), 469-476. doi: 10.1016/j.na.2011.08.048.

[14]

H.-J. Ruan and Y.-W. Zheng, The "hot spots" conjecture on the level-3 Sierpinski gasket, Nonlinear Anal., 81 (2013), 101-109. doi: 10.1016/j.na.2012.10.014.

[15]

T. Shima, On eigenvalue problems for the random walks on the Sierpinski pre-gaskets, Japan J. Indust. Appl. Math., 8 (1991), 127-141. doi: 10.1007/BF03167188.

[16]

T. Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math., 13 (1996), 1-23. doi: 10.1007/BF03167295.

[17]

R. S. Strichartz, Differential Equations on Fractals, Princeton University Press, 2006.

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