- Previous Article
- CPAA Home
- This Issue
-
Next Article
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. |
| [1] |
Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934/dcdss.2011.4.833 |
| [2] |
James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3029-3063. doi: 10.3934/cpaa.2021095 |
| [3] |
Shiping Cao, Hua Qiu. Boundary value problems for harmonic functions on domains in Sierpinski gaskets. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1147-1179. doi: 10.3934/cpaa.2020054 |
| [4] |
D. Kelleher, N. Gupta, M. Margenot, J. Marsh, W. Oakley, A. Teplyaev. Gaps in the spectrum of the Laplacian on $3N$-Gaskets. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2509-2533. doi: 10.3934/cpaa.2015.14.2509 |
| [5] |
Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1 |
| [6] |
Shigehiro Sakata, Yuta Wakasugi. Movement of time-delayed hot spots in Euclidean space for a degenerate initial state. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2705-2738. doi: 10.3934/dcds.2020147 |
| [7] |
Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198 |
| [8] |
Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1 |
| [9] |
Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491 |
| [10] |
Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845 |
| [11] |
Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 |
| [12] |
Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 |
| [13] |
Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130 |
| [14] |
Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074 |
| [15] |
Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201 |
| [16] |
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 |
| [17] |
Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239 |
| [18] |
Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042 |
| [19] |
Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977 |
| [20] |
Yibin Zhang. The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3445-3476. doi: 10.3934/cpaa.2020151 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]