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Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators
The "hot spots" conjecture on higher dimensional Sierpinski gaskets
1. | School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China, China |
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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