Skip to main content
SpringerLink
Log in

Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for carnot-carathéodory metrics

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Bony, J.M. (1969). Principe du maximum, inégalité de Harnack et du problème de Cauchy pour les operateurs elliptique degeneres.Ann. Inst. Fourier, Grenoble,119(1), 277–304.

    MathSciNet  Google Scholar 

  2. Buckley, S., Koskela, P., and Lu, G. (1996). Boman equals John.XVI-th Rolf Nevanlinna Colloquium, Laine/Martio, Eds. Walter de Gruyter, & Co., Berlin, New York, 91–99.

    Google Scholar 

  3. Caffarelli, L., Fabes, E., Mortola, S., and Salsa, S. (1981). Boundary behavior of nonnegative solutions of elliptic operators in divergence form.Indiana Univ. Math. J.,30(4), 621–640.

    Article  MathSciNet  MATH  Google Scholar 

  4. Capogna, L., Danielli, D., and Garofalo, N. (1993). An embedding theorem and the Harnack inequality for nonlinear subelliptic equations.Comm. P.D.E.,18(9–10), 1765–1794.

    MathSciNet  MATH  Google Scholar 

  5. Capogna, L., Danielli, D., and Garofalo, N. (1996). Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations.Amer. J. Math.,118(6), 1153–1196.

    MathSciNet  MATH  Google Scholar 

  6. Capogna, L., Garofalo, N., and Nhieu, D.-M. (1997). The Dirichlet problem for sub-Laplacians. Preprint.

  7. Capogna, L. and Tang, P. (1995). Uniform domains and quasiconformal mappings in the Heisenberg group.Man. Math.,86(3), 267–282.

    MathSciNet  MATH  Google Scholar 

  8. Citti, G. (1988). Wiener estimates at boundary points for Hörmander's operators.Boll. U.M.I.,2-B, 667–681.

    MathSciNet  Google Scholar 

  9. Citti, G., Garofalo, N., and Lanconelli, E. (1993). Harnack inequality for sum of squares of vector fields plus a potential.Amer. J. Math.,115(3), 699–734.

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow, W.L. (1939). Über system von linearen partiellen Differentialgleichungen erster Ordnug.Math. Ann.,117, 98–105.

    Article  MathSciNet  MATH  Google Scholar 

  11. Coifman, R.R. and Weiss, G. (1971). Analyse harmonique non-commutative sur certaines espaces homogènes; étude de certains intégrales singulières.Lecture Notes in Mathematics,242, springer-Verlag, Berlin, New York.

    MATH  Google Scholar 

  12. Dahlberg, B.E.J. (1977). On estimates of harmonic measure.Arch. Rat. Mech. Analysis,65, 272–288.

    Article  MathSciNet  Google Scholar 

  13. Danielli, D. (1995). Regularity at the boundary for solutions of nonlinear subelliptic equations.Indiana Univ. Math. J.,44(1), 269–286.

    Article  MathSciNet  MATH  Google Scholar 

  14. Danielli, D., Garofalo, N., and Nhieu, D.-M. (1996). Trace inequalities for Carnot-Carathéodory spaces and applications. Preprint.

  15. Fichera, G. (1955–1956). Sulle equazioni alle derivate parziali del secondo ordine ellittico-paraboliche.Univ. Pol. Torino, Rend. Sem. Mat.,15, 27–47.

    MathSciNet  Google Scholar 

  16. Fichera, G. (1956). Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine.Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I,18(5), 1–30.

    MathSciNet  Google Scholar 

  17. Fichera, G. (1960). On a unified theory of boundary value problems for elliptic-parabolic equations of second order.Boundary Problems in Differential Equations, Univ. of Wisconsin Press, Madison. 97–120.

    Google Scholar 

  18. Folland, G.B. (1973). A fundamental solution for a subelliptic operator.Bull. Amer. Math. Soc.,79, 373–376.

    Article  MathSciNet  MATH  Google Scholar 

  19. Folland, G.B. (1975). Subelliptic estimates and function spaces on nilpotent Lie groups.Arkiv. für Math.,13, 161–207.

    Article  MathSciNet  MATH  Google Scholar 

  20. Folland, G.B., and Stein, E.M. (1982). Hardy spaces on homogeneous groups.Math. Notes,28, Princeton University Press, Princeton, NJ.

    MATH  Google Scholar 

  21. Garofalo, N., and Nhieu, D.M. (1996). Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces.Comm. Pure and Appl. Math.,49 1081–1144.

    Article  MathSciNet  MATH  Google Scholar 

  22. Garofalo, N. and Nhieu, D.M. (1997). Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces.J. d'Anal. Math.,37.

  23. Gaveau, B. (1977). Principe de moindre action, propagation de la chaleur et estimates souselliptiques sur certaines groupe nilpotents.Acta Math.,139(1–2), 95–152.

    Article  MathSciNet  MATH  Google Scholar 

  24. Gilbarg, D. and Trudinger, N. (1983).Elliptic Partial Differential Equations of Second Order. Springer-Verlag.

  25. Hansen, W. and Huber, H. (1984). The Dirichlet problem for sublaplacians on nilpotent groups—Geometric criteria for regularity.Math. Ann.,246, 537–547.

    Google Scholar 

  26. Hervé, R.M. and Hervé, M. (1972). Les functions surharmoniques dans l'axiomatique de M. Brelot associées à un operateur elliptique dégénéré.Ann. Inst. Fourier, (Grenoble),22(2), 131–145.

    MathSciNet  MATH  Google Scholar 

  27. Hörmander, L. (1967). Hypoelliptic second order differential equations.Acta Math.,119, 147–171.

    Article  MathSciNet  MATH  Google Scholar 

  28. Hunt, R.R., and Wheeden, R.L. (1968). On the boundary values of harmonic functions.Trans. Amer. Math. Soc.,132, 307–322.

    Article  MathSciNet  MATH  Google Scholar 

  29. Hunt, R.R., and Wheeden, R.L. (1970). Positive harmonic functions of Lipschitz domains.Trans. Amer. Math. Soc.,147, 507–527.

    Article  MathSciNet  MATH  Google Scholar 

  30. Huber, H. (1986). Examples of irregular domains for some hypoelliptic differential operators.Expo Math.,4, 189–192.

    Google Scholar 

  31. Jerison, D.S. (1981). The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, Parts I and II.J. Funct. Analysis,43, 97–142.

    Article  MathSciNet  MATH  Google Scholar 

  32. Jerison, D.S. (1983). Boundary regularity in the Dirichlet problem for □ b on CR manifolds.Comm. Pure and Appl. Math.,36, 143–181.

    MathSciNet  MATH  Google Scholar 

  33. Jerison, D.S., and Kenig, C.E. (1982). Boundary behavior of harmonic functions in non-tangentially accessible domains.Adv. in Math.,46, 80–147.

    Article  MathSciNet  MATH  Google Scholar 

  34. Jones, P.W. (1982). A geometric localization theorem.Adv. in Math.,46, 71–79.

    Article  MathSciNet  MATH  Google Scholar 

  35. Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms.Trans. Amer. Math. Soc.,258, 147–153.

    Article  MathSciNet  MATH  Google Scholar 

  36. Kemper, J. (1972). Temperatures in several variables; kernel functions, representations, and parabolic boundary values.Trans. Amer. Math. Soc.,167, 243–262.

    Article  MathSciNet  MATH  Google Scholar 

  37. Kenig, C.E. (1994). Harmonic analysis techniques for second order elliptic boundary value problems.Regional Conference Series in Mathematics,83, AMS.

  38. Kohn, J.J., and Nirenberg, L. (1965). Non-coercive boundary value problems.Comm. Pure and Appl. Math.,18, 443–492.

    MathSciNet  MATH  Google Scholar 

  39. Korányi, A. (1983). Geometric aspects of analysis on the Heisenberg group.Topics in Modern Harmonic Analysis,1, 209–258. 1st. Naz. Alta Mat. F. Severi. Rome.

    Google Scholar 

  40. Korányi, A., and Riemann, H.M. (1995). Foundations for the theory of quasiconformal mappings on the Heisenberg group.Adv. in Math.,111(1), 1–87.

    Article  MATH  Google Scholar 

  41. Lewis, J. Private communication.

  42. Littman, W., Stampacchia, G., and Weinberger, H.F. (1963). Regular points for elliptic equations with discontinuous coefficients.Ann. S. N. S. Pisa,3(17), 43–79.

    MathSciNet  Google Scholar 

  43. Lu, G. (1994). On Harnack's inequality for a class of strongly degenerate Schrödinger operators formed by vector fields.Diff. Int. Eq.,7(1), 73–100.

    MATH  Google Scholar 

  44. Nagel, A., Stein, E.M., and Wainger, S. (1985). Balls and metrics defined by vector fields I: basic properties.Acta Math.,155, 103–147.

    Article  MathSciNet  MATH  Google Scholar 

  45. Negrini, P., and Scornazzani, V. (1987). Wiener criterion for a class of degenerate elliptic operators.J. Diff. Eq.,66, 151–167.

    Article  MathSciNet  MATH  Google Scholar 

  46. Nhieu, D.M. (1994). Extension theorems for Sobolev spaces in stratified nilpotent Lie groups. Preprint.

  47. Oleinik, O.A., and Radkevich, E.V. (1973). Second order equations with non-negative characteristic form. (Mathematical Analysis 1969), Moscow: Itogi Nauki (1971) [Russian], English translation:Amer. Math. Soc., Providence, RI.

    Google Scholar 

  48. Pansu, P. (1989). Métriques de Carnot-Carathéodory et quasisométries des espaces symmetriques de rang un.Ann. Math.,129, 1–60.

    Article  MathSciNet  Google Scholar 

  49. Sanchez-Calle, A. (1984). Fundamental solutions and geometry of sum of squares of vector fields.Inv. Math.,78, 143–160.

    Article  MathSciNet  MATH  Google Scholar 

  50. Varadarajan, V.S. (1974).Lie Groups, Lie Algebras, and Their Representation. Prentice Hall, Englewood Cliffs, NJ

    Google Scholar 

  51. Vodop'yanov, S.K. (1995). Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations.Sib. Math. J.,36(2), 246–264.

    Article  MathSciNet  MATH  Google Scholar 

  52. Wittmann, R. (1987). A nontangential limit theorem.Osaka J. Math.,24, 61–72.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, NY

    Luca Capogna & Nicola Garofalo

  2. Department of Mathematics, Purdue University, 47907, West Lafayette, IN

    Luca Capogna & Nicola Garofalo

  3. Dipartimento di Metodi e Modelli Matematici, Università di Padova, Via Belzoni 7, 35131, Padova, Italy

    Luca Capogna & Nicola Garofalo

Authors
  1. Luca Capogna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicola Garofalo
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A Gene, con nostalgia

Acknowledgments and Notes. Nicola Garofalo-Supported by the NSF, grant DMS-9706892.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Capogna, L., Garofalo, N. Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for carnot-carathéodory metrics. The Journal of Fourier Analysis and Applications 4, 403–432 (1998). https://doi.org/10.1007/BF02498217

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02498217

Keywords

Search

Navigation