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On a class of rotationally symmetric $p$-harmonic maps
| 1. | Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong |
| 2. | Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Taiwan |
| 3. | Department of Mathematics, National University of Singapore, Singapore 119260, Singapore |
We give a classification of rotationally symmetric $p$-harmonic maps between some model spaces such as $\mathbb{R}^n$ and $\mathbb{H}^n$ by their asymptotic behaviors. Among others, we show that, when $p>2$ and $n≥q 2$, all rotationally symmetric $p$-harmonic maps from $\mathbb{R}^n$ to $\mathbb{H}^n$ have to blow up at a finite point, while all rotationally symmetric $p$-harmonic maps from $\mathbb{H}^n$ to $\mathbb{H}^n$ observe the trichotomy property, i.e. the map $y$ is the identity map, is bounded or blows up according as its initial value $y'(0)$ is equal to, less than or greater than one. Our sharp estimates imply and improve a number of existence and non-existence results of certain $p$-harmonic maps on noncompact manifolds.
References:
| [1] |
C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law,
On the blowup of heat flow for conformal 3-harmonic maps, Trans. Amer. Math. Soc., 354 (2002), 5087-5110.
doi: 10.1090/S0002-9947-02-03054-4. |
| [2] |
C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law,
Integrability of rotationally symmetric $n$-harmonic maps, J. Math. Anal. Appl., 327 (2007), 869-877.
doi: 10.1016/j.jmaa.2006.04.073. |
| [3] |
Y. Chen, M. C. Hong and N. Hungerbühler,
Heat flow of p-harmonic maps with values into spheres, Math. Z., 215 (1994), 25-35.
doi: 10.1007/BF02571698. |
| [4] |
L. F. Cheung and C. K. Law,
An initial value approach to rotationally symmetric harmonic maps, J. Math. Anal. Appl., 289 (2004), 1-13.
doi: 10.1016/S0022-247X(03)00195-1. |
| [5] |
L. F. Cheung, C. K. Law and M. C. Leung,
Bounded positive solutions of rotationally symmetric harmonic map equations, Diff. Integral Equations, 13 (2000), 1149-1188.
|
| [6] |
L. F. Cheung, C. K. Law, M. C. Leung and J. B. McLeod,
Entire solutions of quasilinear differential equations corresponding to p-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 701-715.
doi: 10.1016/S0362-546X(97)00434-3. |
| [7] |
R. Dal Passo, L. Giacomelli and S. Moll,
Rotationally symmetric 1-harmonic maps from $D^2$ to $S^2$, Cal. Var. P.D.E., 32 (2008), 533-554.
doi: 10.1007/s00526-007-0153-2. |
| [8] |
A. Fardoun and R. Regbaoui,
Heat flow for $p$-harmoic maps with small initial data, Cal. Var., P.D.E., 16 (2003), 1-16.
doi: 10.1007/s005260100138. |
| [9] |
R. Greene and H. Wu,
Function Theory on Manifolds Which Possess a Pole Lecture Notes in Math. 699 (1970), Springer-Verlag. |
| [10] |
R. G. Iagar and S. Moll,
Rotationally symmetric $p$-harmonic maps from $D^2$ to $S^2$, J. Differential Equations, 254 (2013), 3928-3956.
doi: 10.1016/j.jde.2013.02.003. |
| [11] |
M. C. Leung,
Asymptotic behavior of rotationally symmetric $p$-harmonic maps, Applicable Anal., 61 (1996), 1-15.
doi: 10.1080/00036819608840440. |
| [12] |
M. C. Leung,
Positive solutions of second order quasilinear equations corresponding to $p$-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 717-733.
doi: 10.1016/S0362-546X(97)00435-5. |
| [13] |
M. Misawa,
On the $p$-harmonic flow into spheres in the singular case, Nonlinear Anal., 50 (2002), 485-494.
doi: 10.1016/S0362-546X(01)00755-6. |
| [14] |
A. Ratto and M. Rigoli,
On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Diff. Eqns., 101 (1993), 15--27.
doi: 10.1006/jdeq.1993.1002. |
show all references
References:
| [1] |
C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law,
On the blowup of heat flow for conformal 3-harmonic maps, Trans. Amer. Math. Soc., 354 (2002), 5087-5110.
doi: 10.1090/S0002-9947-02-03054-4. |
| [2] |
C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law,
Integrability of rotationally symmetric $n$-harmonic maps, J. Math. Anal. Appl., 327 (2007), 869-877.
doi: 10.1016/j.jmaa.2006.04.073. |
| [3] |
Y. Chen, M. C. Hong and N. Hungerbühler,
Heat flow of p-harmonic maps with values into spheres, Math. Z., 215 (1994), 25-35.
doi: 10.1007/BF02571698. |
| [4] |
L. F. Cheung and C. K. Law,
An initial value approach to rotationally symmetric harmonic maps, J. Math. Anal. Appl., 289 (2004), 1-13.
doi: 10.1016/S0022-247X(03)00195-1. |
| [5] |
L. F. Cheung, C. K. Law and M. C. Leung,
Bounded positive solutions of rotationally symmetric harmonic map equations, Diff. Integral Equations, 13 (2000), 1149-1188.
|
| [6] |
L. F. Cheung, C. K. Law, M. C. Leung and J. B. McLeod,
Entire solutions of quasilinear differential equations corresponding to p-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 701-715.
doi: 10.1016/S0362-546X(97)00434-3. |
| [7] |
R. Dal Passo, L. Giacomelli and S. Moll,
Rotationally symmetric 1-harmonic maps from $D^2$ to $S^2$, Cal. Var. P.D.E., 32 (2008), 533-554.
doi: 10.1007/s00526-007-0153-2. |
| [8] |
A. Fardoun and R. Regbaoui,
Heat flow for $p$-harmoic maps with small initial data, Cal. Var., P.D.E., 16 (2003), 1-16.
doi: 10.1007/s005260100138. |
| [9] |
R. Greene and H. Wu,
Function Theory on Manifolds Which Possess a Pole Lecture Notes in Math. 699 (1970), Springer-Verlag. |
| [10] |
R. G. Iagar and S. Moll,
Rotationally symmetric $p$-harmonic maps from $D^2$ to $S^2$, J. Differential Equations, 254 (2013), 3928-3956.
doi: 10.1016/j.jde.2013.02.003. |
| [11] |
M. C. Leung,
Asymptotic behavior of rotationally symmetric $p$-harmonic maps, Applicable Anal., 61 (1996), 1-15.
doi: 10.1080/00036819608840440. |
| [12] |
M. C. Leung,
Positive solutions of second order quasilinear equations corresponding to $p$-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 717-733.
doi: 10.1016/S0362-546X(97)00435-5. |
| [13] |
M. Misawa,
On the $p$-harmonic flow into spheres in the singular case, Nonlinear Anal., 50 (2002), 485-494.
doi: 10.1016/S0362-546X(01)00755-6. |
| [14] |
A. Ratto and M. Rigoli,
On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Diff. Eqns., 101 (1993), 15--27.
doi: 10.1006/jdeq.1993.1002. |
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