Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

Abstract Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group ℍ, carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.

[1]  D. Jerison,et al.  The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II , 1981 .

[2]  E. Lanconelli,et al.  Non-existence results for semilinear kohn-laplace equations in unbounded domains , 2000 .

[3]  E. Lanconelli,et al.  A negative answer to a one-dimensional symmetry problem in the Heisenberg group , 2001, math/0108198.

[4]  E. Lanconelli,et al.  Zero-order perturbations of the subelliptic Laplacian on the Heisenberg group and their uniqueness properties , 1990 .

[5]  Igor Leite Freire Noether Symmetries and Conservations Laws For Non-critical Kohn-Laplace Equations on Three-Dimensional Heisenberg Group , 2007, 0706.1745.

[6]  N. Ibragimov Transformation groups applied to mathematical physics , 1984 .

[7]  L. Véron,et al.  Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group , 2000 .

[8]  John M. Lee,et al.  Intrinsic CR normal coordinates and the CR Yamabe problem , 1989 .

[9]  Transformation groups applied to mathematical physics , 1986 .

[10]  P. Olver Applications of lie groups to differential equations , 1986 .

[11]  G. Citti,et al.  Critical semilinear equations on the Heisenberg group: the effect of the topology of the domain , 2001 .

[12]  E. Lanconelli,et al.  Existence and nonexistence results for semilinear equations on the Heisenberg group , 1992 .

[13]  Basilis Gidas,et al.  Global and local behavior of positive solutions of nonlinear elliptic equations , 1981 .

[14]  Sara Maad Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group , 2005 .

[15]  G. Bluman,et al.  Direct Construction of Conservation Laws from Field Equations , 1997 .

[16]  G. Folland Harmonic Analysis in Phase Space. (AM-122), Volume 122 , 1989 .

[17]  E. Lanconelli,et al.  Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation , 1990 .

[18]  Yuri Bozhkov,et al.  Lie Symmetries and Criticality of Semilinear Differential Systems , 2007 .

[19]  E. Stein,et al.  Hypoelliptic differential operators and nilpotent groups , 1976 .

[20]  A. Bonfiglioli,et al.  Nonlinear Liouville theorems for some critical problems on H-type groups , 2004 .

[21]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[22]  Conservation laws of scaling-invariant field equations , 2003, math-ph/0303066.

[23]  E. Stein,et al.  Estimates for the complex and analysis on the heisenberg group , 1974 .

[24]  Thomas Wolf,et al.  A comparison of four approaches to the calculation of conservation laws , 2002, European Journal of Applied Mathematics.

[25]  Group classification of semilinear Kohn–Laplace equations , 2007, math/0703700.

[26]  Divergence symmetries of critical Kohn-Laplace equations on Heisenberg groups , 2007, math/0703698.

[27]  Basilis Gidas,et al.  Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth , 1989 .

[28]  Yuri Bozhkov,et al.  Noether Symmetries and Critical Exponents , 2005 .

[29]  Asymptotics for some green kernels on the Heisenberg group and the Martin boundary , 1989 .

[30]  K. Hannabuss,et al.  HARMONIC ANALYSIS IN PHASE SPACE: (Annals of Mathematics Studies 122) , 1990 .

[31]  I. Dolcetta,et al.  Indefinite semi–linear equations on the heisenberg group: a prior1 bounds and existence , 1998 .

[32]  G. Bluman,et al.  Direct construction method for conservation laws of partial differential equations Part II: General treatment , 2001, European Journal of Applied Mathematics.

[33]  Stephen C. Anco,et al.  Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications , 2001, European Journal of Applied Mathematics.

[34]  John M. Lee,et al.  Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem , 1988 .

[35]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .