Skip to main content
Log in

On some properties of solutions of quasilinear degenerate equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

For quasilinear equations div A(x, u, ∇u) = 0 with degeneracy ω(x) of the Muckenhoupt A p -class, we prove the Harnack inequality, an estimate for the Hölder norm, and a sufficient criterion for the regularity of boundary points of the Wiener type.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Littman, G. Stampacchia, and H. F. Weinherger, “Regular points for equations with discontinuous coefficients,” Ann. Sci. Norm. Super. Pisa Cl. Sci., 17, 43–77 (1963).

    MATH  Google Scholar 

  2. E. B. Fabes, C. E. Kenig, and R. P. Serapioni, “The local regularity of solutions of degenerate elliptic equations,” Commun. Part. Different. Equat., 7, 77–116 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  3. E. B. Fabes, D. Jersion, and C. Kenig, “The Wiener test degenerate elliptic equations,” Ann. Inst. Fourier (Grenoble), 32, 151–182 (1982).

    Google Scholar 

  4. S. Chanillo and R. L. Wheeden, “Harnack's inequality and mean-value inequalities for solutions of degenerated elliptic equations,” Commun. Part. Different. Equat., 11 (10), 1111–1134 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Garieppy and W. P. Ziemer, “A regularity condition at the boundary for solutions of quasilinear elliptic equations,” Arch. Ration. Mech. Anal., 67, 25–39 (1977).

    Article  Google Scholar 

  6. V. A. Kondrat'ev and E. M. Landis, “Qualitative theory of linear partial differential equations of the second order,” in: VINITI Series in Contemporary Problems of Mathematics, Fundamental Trends [in Russian], Vol. 32, VINITI, Moscow (1988), pp. 99–217.

    Google Scholar 

  7. V. G. Maz'ya, “On Wiener's type regularity of a boundary point for higher-order elliptic equations,” in: M. Krbec and A. Kufner (editors), Nonlinear Analysis, Function Spaces and Applications (Proceedings of the Spring School Held in Prague, May 31–June 6, 1988), Vol. 6 (1988), pp. 119–155.

  8. J. Maly and W. P. Ziemer, Regularity of Solutions of Elliptic Partial Differential Equations, American Mathematical Society, Providence, RI (1997).

    MATH  Google Scholar 

  9. E. M. Landis, Second-Order Equations of Elliptic and Parabolic Types [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  10. A. Kufner, Weighted Sobolev Spaces, Feubner, Leipzig (1980).

    MATH  Google Scholar 

  11. T. Kilpelainen, “Smooth approximation in weighted Sobolev spaces,” Comment. Math. Univ. Carol., 38, 29–35 (1997).

    MathSciNet  Google Scholar 

  12. B. Franchi, R. Serapioni, and F. S. Cassano, “Approximation and embedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields,” Boll. Unione Mat. Ital., 7, No. 11-B, 83–117 (1997).

    Google Scholar 

  13. E. T. Sawyer and R. L. Wheeden, “Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces,” Amer. J. Math., 114, 813–874 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  14. F. I. Mamedov, “On two weighted Sobolev inequalities in unbounded domains,” in: V. M. Kokilashvili (editor), Proceedings of the A. Razmadze Mathematical Institute of the Georgian Academy of Sciences, Vol. 21 (1999), pp. 117–123.

  15. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  16. E. M. Landis, “On some problems in the qualitative theory of elliptic equations,” Usp. Mat. Nauk, 18, No. 1, 3–62 (1963).

    MathSciNet  Google Scholar 

  17. V. G. Maz'ya, Sobolev Spaces [in Russian], Leningrad University, Leningrad (1985).

    Google Scholar 

  18. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires [Russian translation], Mir, Moscow (1972).

    MATH  Google Scholar 

  19. P. Drabek, A. Kufner, and F. Nicolosi, Nonlinear Elliptic Equations (Singular and Degenerate Cases), University of West Bohemia, Pilsen (1996).

    Google Scholar 

  20. E. M. Stein, Singular Integrals and Differential Properties of Functions, Princeton University, Princeton (1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 918–936, July, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mamedov, F.I., Amanov, R.A. On some properties of solutions of quasilinear degenerate equations. Ukr Math J 60, 1073–1098 (2008). https://doi.org/10.1007/s11253-008-0108-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-008-0108-6

Keywords

Navigation