Duke Mathematical Journal

Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type

Nicola Garofalo and Dimiter Vassilev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper we study positive solutions to the CR Yamabe equation on groups of Heisenberg type. For the subclass of groups of Iwasawa type, we characterize those solutions that have partial symmetry, that is, those that are invariant with respect to the action of the orthogonal group in the first layer of the Lie algebra.

Article information

Source
Duke Math. J., Volume 106, Number 3 (2001), 411-448.

Dates
First available in Project Euclid: 13 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1092403938

Digital Object Identifier
doi:10.1215/S0012-7094-01-10631-5

Mathematical Reviews number (MathSciNet)
MR1813232

Zentralblatt MATH identifier
1012.35014

Subjects
Primary: 35H20: Subelliptic equations
Secondary: 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72] 35B33: Critical exponents 43A80: Analysis on other specific Lie groups [See also 22Exx]

Citation

Garofalo, Nicola; Vassilev, Dimiter. Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type. Duke Math. J. 106 (2001), no. 3, 411--448. doi:10.1215/S0012-7094-01-10631-5. https://projecteuclid.org/euclid.dmj/1092403938


Export citation

References

  • \lccT. Aubin, Espaces de Sobolev sur les variétés Riemanniennes, Bull. Sci. Math. (2) 100 (1976), 149–173.
  • ––––, Problèms isopérimétriques et espaces de Sobolev, J. Diff. Geom. 11 (1976), 573–598.
  • \lccA. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315.
  • \lccW. Beckner, On the Grushin operator and hyperbolic symmetry, to appear in Proc. Amer. Math. Soc.
  • \lccI. Birindelli and A. Cutri, A semi-linear problem for the Heisenberg Laplacian, Rend. Sem. Mat. Univ. Padova 94 (1995), 137–153.
  • \lccI. Birindelli and J. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Comm. Partial Differential Equations 24 (1999), 1875–1890.
  • \lccJ.-M. Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), 277–304.
  • \lccL. Caffarelli, N. Garofalo, and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), 1457–1474.
  • \lccL. Capogna, D. Danielli, and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), 1153–1196.
  • \lccW. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–\hs622.
  • \lccL. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications, Part I: Basic Theory and Examples, Cambridge Stud. Adv. Math. 18, Cambridge Univ. Press, Cambridge, 1990.
  • \lccM. Cowling, A. H. Dooley, A. Korányi, and F. Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), 1–\hs41.
  • \lccM. Cowling and A. Korányi, “Harmonic analysis on Heisenberg type groups from a geometric viewpoint” in Lie Group Representation, III (College Park, Md., 1982/1983), Lecture Notes in Math. 1077, Springer, Berlin, 1984, 60–100.
  • \lccJ. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), 69–70.
  • \lccE. Damek, Harmonic functions on semidirect extensions of type $H$ nilpotent groups, Trans. Amer. Math. Soc. 290 (1985), 375–384.
  • ––––, A Poisson kernel on Heisenberg type nilpotent groups, Collq. Math. 53 (1987), 239–247.
  • ––––, The geometry of semidirect extensions of a Heisenberg type nilpotent group, Collq. Math. 53 (1987), 255–268.
  • \lccG. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376.
  • ––––, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207.
  • \lccG. B. Folland and E. M. Stein, Estimates for the $\Bar {\partial}_{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522.
  • \lccN. Garofalo and E. Sartori, Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates, Adv. Differential Equations 4 (1999), 137–161.
  • \lccN. Garofalo and D. N. Vassilev, Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians, to appear in Math. Ann.
  • \lccB. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.
  • ––––, “Symmetry of positive solutions of nonlinear elliptic equations in $R_j\sp{n}$” in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7A Academic Press, New York, 1981, 369–\hs402.
  • \lccD. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, New York, 1983.
  • \lccH. Hopf, Differential Geometry in the Large, Lecture Notes in Math. 1000, Springer, Berlin, 1983.
  • \lccH. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
  • \lccD. Jerison and J. M. Lee, “A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds” in Microlocal Analysis (Boulder, Colo., 1983), Contemp. Math. 27, Amer. Math. Soc., Providence, 1984, 57–\hs63.
  • ––––, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167–197.
  • ––––, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1–13.
  • ––––, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), 303–343.
  • \lccA. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153.
  • ––––, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), 127–136.
  • ––––, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35–\hs42.
  • \lccA. Kaplan and F. Ricci, “Harmonic analysis on groups of Heisenberg type” in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math. 992, Springer, Berlin, 1983, 416–\hs435.
  • \lccA. Korányi, Kelvin transforms and harmonic polynomials on the Heisenberg group, J. Funct. Anal. 49 (1982), 177–185.
  • ––––, Geometric properties of Heisenberg-type groups, Adv. in Math. 56 (1985), 28–38.
  • \lccP.-L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Part 1, Rev. Mat. Iberoamericana 1 (1985), 145–201.
  • ––––, The concentration-compactness principle in the calculus of variations, the limit case, Part 2, Rev. Mat. Iberoamericana 1 (1985), 45–121.
  • \lccG. Lu and J. Wei, On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 83–89.
  • \lccJ. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304\hs–318.
  • \lccR. Sperb, Maximum Principles and Their Applications, Math. Sci. Engrg. 157, Academic Press, New York, 1981.
  • \lccG. Talenti, Best constant in the Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.
  • \lccD. N. Vassilev, Ph.D. dissertation, Purdue Univ., West Lafayette, Ind., 2000.
  • \lccH. F. Weinberger, Remark on the preceding paper by Serrin, Arch. Rational Mech. Anal. 43 (1971), 319–320. elliptic equations in $R_j\sp{n}$” in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7A Academic Press, New York, 1981, 369–\hs402.
  • \lccD. Gilbarg \and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, New York, 1983.
  • \lccH. Hopf, Differential Geometry in the Large, Lecture Notes in Math. 1000, Springer, Berlin, 1983.
  • \lccH. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
  • \lccD. Jerison \and J. M. Lee, “A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds” in Microlocal Analysis (Boulder, Colo., 1983), Contemp. Math. 27, Amer. Math. Soc., Providence, 1984, 57–\hs63.
  • ––––, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167–197.
  • ––––, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1–13.
  • ––––, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), 303–343.
  • \lccA. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153.
  • ––––, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), 127–136.
  • ––––, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35–\hs42.
  • \lccA. Kaplan \and F. Ricci, “Harmonic analysis on groups of Heisenberg type” in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math. 992, Springer, Berlin, 1983, 416–\hs435.
  • \lccA. Korányi, Kelvin transforms and harmonic polynomials on the Heisenberg group, J. Funct. Anal. 49 (1982), 177–185.
  • ––––, Geometric properties of Heisenberg-type groups, Adv. in Math. 56 (1985), 28–38.
  • \lccP.-L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Part 1, Rev. Mat. Iberoamericana 1 (1985), 145–201.
  • ––––, The concentration-compactness principle in the calculus of variations, the limit case, Part 2, Rev. Mat. Iberoamericana 1 (1985), 45–121.
  • \lccG. Lu \and J. Wei, On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 83–89.
  • \lccJ. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304\hs–318.
  • \lccR. Sperb, Maximum Principles and Their Applications, Math. Sci. Engrg. 157, Academic Press, New York, 1981.
  • \lccG. Talenti, Best constant in the Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.
  • \lccD. N. Vassilev, Ph.D. dissertation, Purdue Univ., West Lafayette, Ind., 2000.
  • \lccH. F. Weinberger, Remark on the preceding paper by Serrin, Arch. Rational Mech. Anal. 43 (1971), 319–320.