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Stability of the equator map for the Hessian energy

Author(s): Min-Chun Hong; Bevan Thompson
Journal: Proc. Amer. Math. Soc. 135 (2007), 3163-3170.
MSC (2000): Primary 35J50
Posted: June 19, 2007
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Abstract: In this paper we show that the equator map is a minimizer of the Hessian energy $ H(u)=\int _{\Omega } \vert\bigtriangleup u\vert^{2}\,dx$ in $ H^{2}(\Omega ;S^{n})$ for $ n\geq 10$ and is unstable for $ 5\le n\le 9.$

References:

[A]
G. Angelsberg, A monotonicity formula for stationary biharmonic maps, Math. Z. 252 (2006), 287 - 293. MR 2207798 (2006k:58022)

[CWY]
S. Y. A. Chang, L. Wang and P. Yang, A regularity theory of biharmonic maps, Comm. Pure Appl. Math. 52 (1999), 1113-1137. MR 1692148 (2000j:58025)

[H]
M.-C. Hong, On the Jäger Kaul theorem concerning harmonic maps, Ann. Inst. H. Poincare, Analyse non Linéare 17 (2000), 35-46. MR 1743430 (2000k:58021)

[HW]
M.-C. Hong, C. Y. Wang, Regularity and relaxed problems of minimizing biharmonic maps into spheres, Calc. Var. Partial Differential Equations 23 (2005), 425-450. MR 2153032 (2006e:49082)

[JK2]
W. Jäger and H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew, Math. 343 (1983), 146-161. MR 705882 (85f:58031)

[W1]
C. Y. Wang, Remarks on biharmonic maps into spheres, PDE Calc. Var. Partial Differential Equations 21 (2004), 221-242. MR 2094320 (2005e:58026)

[W2]
C. Y. Wang, Biharmonic maps form $ \mathbb{R}^{4}$ into a Riemannian manifold, Math. Z. 247 (2004), 65-87. MR 2054520 (2005c:58030)

[W3]
C. Y. Wang, Stationary biharmonic maps form $ \mathbb{R}^{m}$ into a Riemannian manifold, Comm. Pure Appl. Math. LVII (2004), 0419-0444.

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Additional Information:

Min-Chun Hong
Affiliation: Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia
Email: hong@maths.uq.edu.au

Bevan Thompson
Affiliation: Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia
Email: hbt@maths.uq.edu.au

DOI: 10.1090/S0002-9939-07-08950-2
PII: S 0002-9939(07)08950-2
Keywords: Biharmonic maps, Hessian energy, Dirichlet energy, minimizer, unstable
Received by editor(s): May 12, 2006
Posted: June 19, 2007
Additional Notes: The first author acknowledges the support of the Australian Research Council Discovery Grant DP0450140
Communicated by: Chuu-Lian Terng
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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