On a fourth order equation in 3-D

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Xu.dvi ESAIM: Control, Optimisation and Calculus of Variations June 2002, Vol. 8, 1029–1042 URL: http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002023 ON A FOURTH ORDER EQUATION IN 3-D Xingwang Xu ,∗1 and Paul C. Yang ,†2 Abstract. 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. Mathematics Subject Classification. 53C21, 35G20. Received January 14, 2002. 1. Introduction In the analytic study of conformal structures in dimensions greater than two, it is fruitful to consider the family of Q-curvature equations as natural generalization of the Yamabe equation. Since the work of Paneitz [9] there has been a number of such equations introduced by Branson [2] and Fe�erman and Graham [7]. In a series of papers [3, 4] it is demonstrated that solutions of these equations lead to signi�cant results for nonlinear analysis as well as for conformal geometry in dimension four. A number of authors have investigated these equations in dimensions higher than four, for example Djadli et al. [6], Hebey and Robert [8] and Ahmedou et al. [1]. In this paper, we call attention to the validity of the equation in dimension three and begin a preliminary investigation of the fourth order Paneitz equation in the most favorable situation. Let us recall the Paneitz operator P = (−�)2 + δ ( 5 4 Rg − 4Ric ) d− 1 2 Q, (1.1) where Q = −2jRicj2 + 23 32 R2 − 1 4 �R. (1.2) Under a conformal change of metrics �g = u−4g with u > 0, the Paneitz operator enjoys the following conformal covariance property: �Pw = u7 P (uw). (1.3) Keywords and phrases: Paneitz operator, conformal invariance, Sobolev inequality, connected sum. 1 Department of Mathematics, National University of Singapore, 2 Science Drive 2, 119260 Singapore; e-mail: [email protected] ∗ Research of Xu supported by NUS Research Gran

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