ESAIM: Control, Optimisation and Calculus of Variations
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 10 / Issue 02 / April 2004, pp 211-223
- © EDP Sciences, SMAI, 2004
- Published online by Cambridge University Press: 22 February 2011
- DOI: http://dx.doi.org/10.1051/cocv:2004002 (About DOI)
Research Article
On the Paneitz energy on standard three sphere
Yang, Paula1 and Zhu, Meijuna2
a1 Mathematics Department, Princeton University, Princeton, NJ 08544, USA.
a2 Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA; mzhu@math.ou.edu.
Abstract
We prove that the Paneitz energy on the standard three-sphere S 3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.
(Received April 21 2003)
(Revised July 21 2003)
(Online publication March 15 2004)
Key Words:
- Paneitz operator;
- symmetrization;
- extremal metric.
Mathematics Subject Classification:
- 58E11;
- 35G99
- [1] A. Jun , C. Kai-Seng and W. Juncheng , Self-similar solutions for the anisotropic affine curve shortening problem. Calc. Var. Partial Differ. Equ. 13 (2001) 311-337. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [2] T. Branson , Differential operators canonically associated to a conformal structure. Math. Scand. 57 (1985) 293-345. [OpenURL Query Data] [MathSciNet] [Google Scholar]
- [3] Y.S. Choi and X. Xu, Nonlinear biharmonic equation with negative exponent. Preprint (1999).
- [4] Z. Djadli , E. Hebey and M. Ledoux , Paneitz type operators and applications. Duke Math. J. 104 (2000) 129-169. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [5] C. Fefferman and R. Graham, Conformal Invariants, in Élie Cartan et les Mathématiques d'aujourd'hui, Asterisque (1985) 95-116.
- [6] E. Hebey and F. Robert , Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients. Calc. Var. Partial Differ. Equ. 13 (2001) 491-517. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [7] E. Hebey , Sharp Sobolev inequalities of second order. J. Geom. Anal. 13 (2003) 145-162. [OpenURL Query Data] [MathSciNet] [Google Scholar]
- [8] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983).
- [9] G. Talenti , Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1976) 697-718. [OpenURL Query Data] [Google Scholar]
- [10] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations. Math. Ann. 313 (1999) 207-228. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [11] X. Xu and P. Yang , On a fourth order equation in 3-D, A tribute to J.L. Lions. ESAIM: COCV 8 (2002) 1029-1042. [OpenURL Query Data] [CrossRef] [Google Scholar]
Powered by:
