April  2001, 7(2): 329-342. doi: 10.3934/dcds.2001.7.329

Positivity of Paneitz operators

1. 

Department of Mathematics, National University of Singapore, Singapore 119260, Singapore

2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States

Revised  October 2000 Published  January 2001

In this paper, we give two results concerning the positivity property of the Paneitz operator-- a fourth order conformally covariant elliptic operator. We prove that the Paneitz operator is positive for a compact Riemannian manifold without boundary of dimension at least six if it has positve scalar curvature as well as nonnegative $Q-$curvature. We also show that the positivity of the Paneitz operator is preserved in dimensions greater than four in taking a connected sum.
Citation: Xingwang Xu, Paul C. Yang. Positivity of Paneitz operators. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 329-342. doi: 10.3934/dcds.2001.7.329
[1]

Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1885-1905. doi: 10.3934/jimo.2019034

[2]

Shuangjie Peng, Jing Zhou. Concentration of solutions for a Paneitz type problem. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1055-1072. doi: 10.3934/dcds.2010.26.1055

[3]

Emmanuel Hebey and Frederic Robert. Compactness and global estimates for the geometric Paneitz equation in high dimensions. Electronic Research Announcements, 2004, 10: 135-141.

[4]

Giuseppe Buttazzo, Filippo Santambrogio. Asymptotical compliance optimization for connected networks. Networks & Heterogeneous Media, 2007, 2 (4) : 761-777. doi: 10.3934/nhm.2007.2.761

[5]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235

[6]

Anca Radulescu. The connected Isentropes conjecture in a space of quartic polynomials. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 139-175. doi: 10.3934/dcds.2007.19.139

[7]

Anna Karnauhova, Stefan Liebscher. Connected components of meanders: Ⅰ. bi-rainbow meanders. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4835-4856. doi: 10.3934/dcds.2017208

[8]

G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485

[9]

Litao Guo, Bernard L. S. Lin. Vulnerability of super connected split graphs and bisplit graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1179-1185. doi: 10.3934/dcdss.2019081

[10]

Phil Howlett, Julia Piantadosi, Paraskevi Thomas. Management of water storage in two connected dams. Journal of Industrial & Management Optimization, 2007, 3 (2) : 279-292. doi: 10.3934/jimo.2007.3.279

[11]

Kaïs Ammari, Denis Mercier. Boundary feedback stabilization of a chain of serially connected strings. Evolution Equations & Control Theory, 2015, 4 (1) : 1-19. doi: 10.3934/eect.2015.4.1

[12]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[13]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39

[14]

Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003

[15]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155

[16]

Ariel Salort. Lower bounds for Orlicz eigenvalues. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021158

[17]

Robert Denk, Leonid Volevich. A new class of parabolic problems connected with Newton's polygon. Conference Publications, 2007, 2007 (Special) : 294-303. doi: 10.3934/proc.2007.2007.294

[18]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709

[19]

Mohamed Badreddine, Thomas K. DeLillo, Saman Sahraei. A Comparison of some numerical conformal mapping methods for simply and multiply connected domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 55-82. doi: 10.3934/dcdsb.2018100

[20]

Manuel Fernández-Martínez. A real attractor non admitting a connected feasible open set. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 723-725. doi: 10.3934/dcdss.2019046

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (131)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]