Compactness and global estimates for the geometric Paneitz equation in high dimensions
Authors:
Emmanuel Hebey and Frédéric Robert
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 135-141
MSC (2000):
Primary :, 58E30, 58J05
DOI:
https://doi.org/10.1090/S1079-6762-04-00138-6
Published electronically:
December 10, 2004
MathSciNet review:
2119034
Full-text PDF Free Access
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Abstract: Given $(M,g)$, a smooth compact Riemannian manifold of dimension $n \ge 5$, we investigate compactness for the fourth order geometric equation $P_gu = u^{2^\sharp -1}$, where $P_g$ is the Paneitz operator, and $2^\sharp = 2n/(n-4)$ is critical from the Sobolev viewpoint. We prove that the equation is compact when the Paneitz operator is of strong positive type.
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HebRobWen E. Hebey, F. Robert, and Y. Wen, Compactness and global estimates for a fourth order equation of critical Sobolev growth arising from conformal geometry, Preprint of the University of Nice, 697, 2004.
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AgmDouNir1 S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727.
AgmDouNir2 S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92.
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ChaYan S.Y.A. Chang and P.C. Yang, On a fourth order curvature invariant, Comp. Math. 237, Spectral Problems in Geometry and Arithmetic, Ed: T. Branson, AMS, 9–28, 1999.
EspRob P. Esposito and F. Robert, Mountain pass critical points for Paneitz-Branson operators, Calc. Var. Partial Differential Equations 15 (2002), 493–517.
HebRob E. Hebey and F. Robert, Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations 13 (2001), 491–517.
HebRobWen E. Hebey, F. Robert, and Y. Wen, Compactness and global estimates for a fourth order equation of critical Sobolev growth arising from conformal geometry, Preprint of the University of Nice, 697, 2004.
Pan S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983.
Sch R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math. 1365, Springer-Verlag, Berlin, 1989, pp. 120–154.
SchYau R. Schoen and S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47–71.
XuYan X. Xu and P. Yang, Positivity of Paneitz operators, Disc. Cont. Dynamical Systems 7 (2001), 329–342.
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Additional Information
Emmanuel Hebey
Affiliation:
Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Email:
Emmanuel.Hebey@math.u-cergy.fr
Frédéric Robert
Affiliation:
Laboratoire J.A.Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France
Email:
frobert@math.unice.fr
Keywords:
Blow-up behavior,
compactness,
Paneitz operator
Received by editor(s):
October 7, 2004
Published electronically:
December 10, 2004
Communicated by:
Tobias Colding
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
