Advertisement

Geometric & Functional Analysis GAFA

, Volume 5, Issue 6, pp 924–947 | Cite as

Conformally invariant systems of nonlinear PDE of Liouville type

  • S. Chanillo
  • M. K. -H. Kiessling
Article

Abstract

We establish a strict isoperimetric inequality and a Pohozaev-Rellich identity for the system
iI = {1,...,N} under certain reasonable conditions on the γιJ anduι. Thus we prove that under these conditions, all solutionsuι are radial symmetric and decreasing about some point.

Keywords

Invariant System Isoperimetric Inequality Reasonable Condition Liouville Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, 1980.Google Scholar
  2. [Be]W. Beckner, to appear.Google Scholar
  3. [Ben]W. H. Bennett, Magnetically self-focussing streams, Phys. Rev. 45 (1934), 890–897Google Scholar
  4. [BrMe]H. Brezis, F. Merle, Uniform estimates and blow-up behavior of solutions of −Δu=V(x)e u in two dimensions. Commun. PDE 16 (1991), 1223–1253.Google Scholar
  5. [CLMP]E. Caglioti, P.-L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanical description, Commun. Math. Phys. 143 (1992), 501–525.Google Scholar
  6. [CaLo]E.A. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality, and Onofri's inequality onS n, Geom. Funct. Anal. 2 (1992), 90–104.Google Scholar
  7. [ChKi]S. Chanillo, M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Commun. Math. Phys. 160:2 (1994), 217–238.Google Scholar
  8. [ChLi]S. Chanillo, Y.Y. Li, Continuity of solutions of uniformly elliptic equations in ℝ2, Manuscr. Math. 77 (1992), 415–433.Google Scholar
  9. [CheLi]W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622.Google Scholar
  10. [ChoW]K.S. Chou, T.Y.H. Wan, Asymptotic radial symmetry for solutions of Δu+e u=0 in a punctured disc, Pac. J. Math. 163 (1994), 269–276.Google Scholar
  11. [ESp]G. Eyink, H. Spohn, Negative temperature states and large-scale long-lived vortices in two-dimensional turbulence, J. Stat. Phys. 70 (1993), 833–886.Google Scholar
  12. [GNNi]B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979), 209–243.Google Scholar
  13. [GiT]D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, New York (1983).Google Scholar
  14. [KP]C. Kesavan, F. Pacella, Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities, Appl. Anal. 54 (1994), 27–37.Google Scholar
  15. [Ki]M.K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math. 46 (1993), 27–56.Google Scholar
  16. [KiL]M.K.-H. Kiessling, J.L. Lebowitz, Dissipative Stationary Plasmas: Kinetic Modeling, Bennett's Pinch, and generalizations, Phys. Plasmas1 (1994), 1841–1849.Google Scholar
  17. [Li]C.-M. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. PDE 16 (1991), 585–615.Google Scholar
  18. [Lio]P.-L. Lions, Two geometrical properties of solutions of semilinear problems, Appl. Anal. 12 (1981), 267–272.Google Scholar
  19. [Liou]J. Liouville, Sur l'equation aux différences partielles ∂2logλ/∂uv±∂/2a 2 = 0, J. Math. 18 (1853), 71–72.Google Scholar
  20. [O]M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom. 6 (1971), 247–258.Google Scholar
  21. [On]E. Onofri, On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys. 86 (1982), 321–326.Google Scholar
  22. [OsPhS]B. Osgood, R. Phillips, P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.Google Scholar
  23. [Po]S.I. Pohozaev, Eigenfunctions of the equation Δuf(u)=0. Sov. Math. Dokl. 5 (1965), 1408–1411.Google Scholar
  24. [R]F. Rellich, Darstellung der Eigenwerte von Δuu=0 durch ein Randwertintegral, Math. Z. 46 (1940), 635–636.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • S. Chanillo
    • 1
  • M. K. -H. Kiessling
    • 2
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Department of MathematicsRutgers UniversityUSA

Personalised recommendations