On a fourth order equation in 3-D
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 1029-1042.

In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.

DOI : 10.1051/cocv:2002023
Classification : 53C21, 35G20
Mots clés : Paneitz operator, conformal invariance, Sobolev inequality, connected sum
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     title = {On a fourth order equation in {3-D}},
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Xu, Xingwang; Yang, Paul C. On a fourth order equation in 3-D. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 1029-1042. doi : 10.1051/cocv:2002023. http://www.numdam.org/articles/10.1051/cocv:2002023/

[1] M. Ahmedou, Z. Djadli and A. Malchiodi, Prescribing a fourth order conformal invariant on the standard sphere. Part I: Perturbation Result. Comm. Comtemporary Math. (to appear). | Zbl

[2] T. Branson, Differential operators cannonically associated to a conformal structure. Math. Scand. 57 (1985) 293-345. | MR | Zbl

[3] A. Chang and P. Yang, Extremal metrics of zeta functional determinants on 4-manifolds. Ann. Math. 142 (1995) 171-212. | MR | Zbl

[4] A. Chang, M. Gursky and P. Yang, An equation of Monge-Ampere type in conformal geometry and four-manifolds of positive Ricci curvature. Ann. Math. (to appear) | Zbl

[5] Y.S. Choi and X. Xu, Nonlinear biharmonic equation with negative exponent. Preprint (1999).

[6] Z. Djadli, E. Hebey and M. Ledoux, Paneitz operators and applications. Duke Math. J. 104 (2000) 129-169. | MR | Zbl

[7] C. Fefferman and R. Graham, Conformal Invariants, in Élie Cartan et les Mathématiques d'aujourd'hui. Asterisque (1985) 95-116. | Numdam | Zbl

[8] E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients. Preprint. | Zbl

[9] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983).

[10] X. Xu and P. Yang, Positivity of Paneitz operators. Discrete Continuous Dynam. Syst. 7 (2001) 329-342. | MR | Zbl

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