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Convex functions on the Heisenberg group

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Abstract.

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.

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References

  1. Alvarez, O., Lasry, J. M., Lions, P. L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. 9 (76), 265-288 (1997)

    Google Scholar 

  2. Ambrosio, L., Magnani, V.: Some fine properties of BV functions on Sub-Riemannian groups. Preprint n. 2, Scuola Normale Superiore, Pisa, Febbraio 2002

  3. Bellaï che, A., Risler, J.-J. (eds.): Sub-Riemannian geometry. Progress in Mathematics, Vol. 144, Birkhäuser 1996

  4. Bieske, T.: On \(\infty-\)harmonic functions on the Heisenberg group. Comm. in Partial Differential Equations 27, 727-762 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cabré, X.: Personal communication

  6. Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations. AMS colloquium publications, Vol. 43. AMS, Providence, RI 1995

  7. Crandall, M.: Viscosity solutions: a primer. Lecture Notes in Mathematics 1660, pp. 1-43. Springer, Berlin Heidelberg New York 1997

  8. Crandall, M., Evans C., Lions, P. L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. of Amer. Math. Soc. 282, 487-502, (1984)

    MathSciNet  MATH  Google Scholar 

  9. Crandall, M., Ishii, H., Lions, P. L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. of Amer. Math. Soc. 27, 1-67 (1992)

    MathSciNet  MATH  Google Scholar 

  10. Danielli, D., Garofalo, N., Nhieu, D.: Notions of convexity in Carnot groups. Preprint

  11. Evans, L. C.: Estimates for smooth absolutely minimizing Lipschitz extensions. Electronic Journal of Differential Equations 3, 1-10 (1993)

    Google Scholar 

  12. Evans, L. C.: Partial differential equations. Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI 1998

  13. Evans, C., Gariepy, R.: Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press LLC 1992

  14. Folland, G. B., Stein, E. M.: Hardy spaces on homogeneous groups. Princeton University Press, Princeton, NJ 1982

  15. Gaveau, B.: Principle de moindre action, propagation de la chaleur, et estimées souselliptiques sur certain groups nilpotents. Acta Mathematica 139, 95-153 (1977)

    MATH  Google Scholar 

  16. Gutierrez, C.: The Monge-Ampére equation. Progress in nonlinear differential equations and their applications 44. Birkhäuser, Boston, MA 2001

  17. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford University Press, New York 1993

  18. Jensen, R.: The maximum principle for viscosity solutions of second order fully nonlinear partial differential equations. Arch. Rat. Mech. Analysis 101, 1-27 (1988)

    MathSciNet  Google Scholar 

  19. Jensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123, 51-74 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Juutinen, P., Lindqvist, P., Manfredi, J.: On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. of Math. Anal. 33, 699-717 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Lanconelli, E., Uguzzoni, F.: On the Poisson kernel for the Kohn Laplacian. Rend. Mat. Appl. (7) 17 (1997), 659-677 (1998)

  22. Lindqvist, P., Manfredi, J.: The Harnack inequality for \(\infty\)-harmonic functions. Electronic Journal of Differential Equations 4, (1995) approx. 5pp. (electronic)

    Google Scholar 

  23. Lindqvist, P., Manfredi, J.: Note on \(\infty\)-harmonic functions. Revista Matemática de la Universidad Complutense 10, 471-480 (1997)

    MATH  Google Scholar 

  24. Lindqvist, P., Manfredi, J., Saksman, E.: Superharmonicity of nonlinear ground states. Rev. Mat. Iberoamericana 16, 17-28 (2000)

    MathSciNet  MATH  Google Scholar 

  25. Lu, G.: Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publicacions Matemátiques 40, 301-329 (1996)

    MATH  Google Scholar 

  26. Lu, G.: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. Revista Matematica Iberoamericana 8, 367-439 (1992)

    MathSciNet  MATH  Google Scholar 

  27. Manfredi, J.: Fully nonlinear subelliptic equations. Available at http://www.pitt.edu/~manfredi/quasi.html

  28. Manfredi, J., Stroffolini B.: A version of the Hopf-Lax Formula in the Heisenberg group. Comm. in Partial Differential Equations 27, 1139-1159 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Monti, R., Serra Cassano, F.: Surface measures in Carnot-Carathéodory spaces. Calc. Var. 13, 339-376 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nagel, A., Stein, E., Wainger, S.: Balls and metrics defined by vector fields I: Basic Properties. Acta Mathematica 55, 103-147 (1985)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, MI 48202, Detroit, USA

    Guozhen Lu, Juan J. Manfredi & Bianca Stroffolini

Authors
  1. Guozhen Lu
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  2. Juan J. Manfredi
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  3. Bianca Stroffolini
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Corresponding author

Correspondence to Guozhen Lu.

Additional information

Received: 5 July 2002, Accepted: 24 October 2002, Published online: 6 June 2003

Mathematics Subject Classification (1991):

49L25, 35J70, 35J67, 22E30

Guozhen Lu: First author supported by US NSF grant DMS-9970352

Juan J. Manfredi: Second author supported by US NSF grant DMS-0100107

Bianca Stroffolini: Third author was supported by G.N.A.M.P.A. and by the 2002 project”Partial Differential Equations and Control Theory”

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Lu, G., Manfredi, J.J. & Stroffolini, B. Convex functions on the Heisenberg group. Cal Var 19, 1–22 (2003). https://doi.org/10.1007/s00526-003-0190-4

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  • DOI: https://doi.org/10.1007/s00526-003-0190-4

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