Skip to main content
SpringerLink
Log in

Exceptional sets for solutions to subelliptic equations

  • Published:
Siberian Mathematical Journal 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.

Institutional subscriptions

References

  1. L. Carleson, Selected Problems on Exceptional Sets [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  2. I. Serrin, “Local behavior of solutions of quasi-linear equations,” Acta Math.,111, No. 3/4, 247–302 (1964).

    Google Scholar 

  3. D. G. Aronson, “Removable singularities for linear parabolic equations,” Arch. Rational Mech. Anal.,17, No. 1, 79–84 (1964).

    Article  Google Scholar 

  4. W. Littman, “Polar sets and removable singularities of partial differential equations,” Ark. Math.,7, No. 1, 1–9 (1965).

    Google Scholar 

  5. R. Harvey and F. Polking, “Removable singularities of solutions of linear partial differential equations,” Acta Math.,125, No. 1/2, 39–56 (1970).

    Google Scholar 

  6. V. G. Maz'ya, “On removable singularities of bounded solutions of quasilinear elliptic equations of any order,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),27, 116–130 (1972).

    Google Scholar 

  7. Yu. V. Egorov, “On removable singularities in boundary conditions for solutions of linear partial differential equations,” Dokl. Akad. Nauk SSSR,289, No. 1, 27–29 (1986).

    Google Scholar 

  8. D. R. Adams, “L p potential theory techniques and nonlinear PDE,” in: Potential Theory: Proceedings of the International Conference (Nagoya, Japan, August–September, 1990), Walter de Gruyter, Berlin-New York, 1992, pp. 1–15.

    Google Scholar 

  9. M. S. Alborova and S. K. Vodop'yanov, “On removable singularities of bounded solutions to quasielliptic equations,” Sibirsk. Mat. Zh.,33, No. 4, 3–14 (1992).

    Google Scholar 

  10. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford-New York-Tokyo (1993).

    Google Scholar 

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

    Google Scholar 

  12. G. Lu, “Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications,” Rev. Mat. Iberoamericana,8, No. 3, 367–439 (1992).

    Google Scholar 

  13. C. J. Xu, “Regularity for quasilinear second-order subelliptic equations,” Comm. Pure Appl. Math.,45, 77–96 (1992).

    Google Scholar 

  14. L. Capogna, D. Danielli, and N. Garofalo, “Embedding theorems and Harnack inequality for solutions of nonlinear subelliptic equations,” C. R. Acad. Sci. Ser. I,316, 809–814 (1993).

    Google Scholar 

  15. S. K. Vodop'yanov, “Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations,” Sibirsk. Mat. Zh.,36, No. 3, 278–300 (1995).

    Google Scholar 

  16. A. Nagel, E. M. Stein, and S. Wainger, “Balls and metrics defined by vector fields. I: Basic properties,” Acta Math.,155, No. 1–2, 103–147 (1985).

    Google Scholar 

  17. B. Franchi, S. Gallot, and R. Wheeden, “Inéqualités isopérimetriques pour des métriques dégénerees,” C. R. Acad. Sci. Ser. I,317, 651–654 (1993).

    Google Scholar 

  18. D. Jerison, “The Poincaré inequality for vector fields satisfying Hörmander's condition,” Duke Math. J.,53, No. 2, 503–523 (1986).

    Article  Google Scholar 

  19. S. K. Vodop'yanov, “L p -potential theory and quasiconformal mappings on homogeneous groups,” in: Contemporary Problems of Geometry and Analysis [in Russian], Trudy Inst. Mat. Vol. 14 (Novosibirsk), Nauka, Novosibirsk, 1989, pp. 45–89.

    Google Scholar 

  20. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Acad. Press, New York (1980).

    Google Scholar 

  21. S. K. Vodop'yanov, “WeightedL p -potential theory on homogeneous groups,” Sibirsk. Mat. Zh.,33, No. 2, 29–48 (1992).

    Google Scholar 

  22. A. Koranyi and S. Vagi, “Singular integrals on homogeneous spaces and some problems of classical analysis,” Ann. Scuola Norm. Sup.,25, 575–648 (1971).

    Google Scholar 

  23. S. K. Vodop'yanov and V. M. Chernikov, “Sobolev spaces and hypoelliptic equations,” in: Trudy Inst. Mat. Vol. 29, Inst. Mat. (Novosibirsk), Novosibirsk, 1995, pp. 3–64.

    Google Scholar 

  24. Yu. G. Reshetnyak, “General theorems on semi continuity and convergence with a functional,” Sibirsk. Mat. Zh.,8, No. 5, 1051–1069 (1967).

    Google Scholar 

  25. Yu. G. Reshetnyak, “On the set of singular points of solutions to certain nonlinear equations of elliptic type,” Sibirsk. Mat. Zh.,9, No. 2, 354–357 (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk

    S. K. Vodop'yanov & I. G. Markina

Authors
  1. S. K. Vodop'yanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. G. Markina
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research was supported by the State Committee for Higher Education of the Russian Federation (Grant 94-1.2-134 in the section of Fundamental Natural Sciences).

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 4, pp. 805–818, July–August, 1995.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vodop'yanov, S.K., Markina, I.G. Exceptional sets for solutions to subelliptic equations. Sib Math J 36, 694–706 (1995). https://doi.org/10.1007/BF02107326

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02107326

Keywords

Search

Navigation