Archive for Rational Mechanics and Analysis

, Volume 216, Issue 2, pp 673–702 | Cite as

Generalized Harnack Inequality for Nonhomogeneous Elliptic Equations

  • Vesa Julin


This paper is concerned with nonlinear elliptic equations in nondivergence form
$$F(D^{2}u, Du, x) = 0 $$
where F has a drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack inequality. This paper presents a new generalization of the Harnack inequality for such equations. As a corollary we obtain the optimal Harnack type of inequality for p(x)-harmonic functions which quantifies the strong minimum principle.


Harmonic Function Elliptic Equation Universal Constant Harnack Inequality Nonnegative Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acerbi E., Mingione G.: Regularity results for a class of functions with non-standard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alarcón, S., García-Melián, J., Quaas, A.: Keller–Osserman type conditions for some elliptic problems with gradient terms. J. Differ. Equ. 252, 886–914 (2012)Google Scholar
  3. 3.
    Alkhutov, Y.A.: The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth conditions. Differ. Uravn. 33, 1651–1660 (1997)Google Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)Google Scholar
  5. 5.
    Armstrong, S.N., Smart, C.K.: Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity. Ann. Probab. 42, 2558–2594 (2014)Google Scholar
  6. 6.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge (1989)Google Scholar
  7. 7.
    Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130, 189–213 (1989)Google Scholar
  8. 8.
    Caffarelli, L.A., Cabre, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI (1995)Google Scholar
  9. 9.
    Fan, X.L., Zhau, D., Zhang, Q.H.: A strong maximum principle for p(x)-Laplace equations. Chin. J. Contemp. Math. 24, 277–282 (2003)Google Scholar
  10. 10.
    Felmer, P., Quaas, A., Sirakov, B.: Solvability of nonlinear elliptic equations with gradient terms. J. Differ. Equ. 254, 4327–4346 (2013)Google Scholar
  11. 11.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)Google Scholar
  12. 12.
    Harjulehto, P., Hästö, P., Latvala, V., Toivanen, O.: The strong minimum principle for quasisuperminimizers of non-standard growth. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 731–742 (2011)Google Scholar
  13. 13.
    Harjulehto, P., Kuusi, T., Lukkari, T., Marola, N., Parviainen, M.: Harnack’s inequality for quasiminimizers with nonstandard growth conditions. J. Math. Anal. Appl. 344, 504–520 (2008)Google Scholar
  14. 14.
    Imbert, C., Silvestre, L.: Estimates on elliptic equations that hold only where the gradient is large. Preprint, 2013. To appear in Journal of the European Mathematical SocietyGoogle Scholar
  15. 15.
    Juutinen, P., Lukkari, T., Parviainen, M.: Equivalence of viscosity and weak solutions for the p(x)-Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 1471–1487 (2010)Google Scholar
  16. 16.
    Koike, S., Takahashi, T.: Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients. Adv. Differ. Equ. 7, 493–512 (2002)Google Scholar
  17. 17.
    Krylov, N.V., Safonov, M.V.: An estimate of the probability that a diffusion process hits a set of positive measure. Dokl. Akad. Nauk. SSSR 245, 253–255, 1979. English translation in Soviet Math. Dokl. 20, 253–255 (1979)Google Scholar
  18. 18.
    Krylov, N.V., Safonov, M.V.: Certain properties of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR 40, 161–175, 1980. English translation in Math. SSSR Izv 161, 151–164 (1980)Google Scholar
  19. 19.
    Savin, O.: Small perturbation solutions for elliptic equations. Commun. Partial Differ. Equ. 32, 557–578 (2007)Google Scholar
  20. 20.
    Sirakov, B.: Solvability of uniformly elliptic fully nonlinear PDE. Arch. Ration. Mech. Anal. 195, 579–607 (2010)Google Scholar
  21. 21.
    Trudinger, N.S.: Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations. Invent. Math. 61, 67–79 (1980)Google Scholar
  22. 22.
    Trudinger N.S.: Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations. Rev. Mat. Iberoamericana 4, 453–468 (1988)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Wolanski, N.: Local bounds, Harnack inequality and Hölder continuity for divergence type elliptic equations with non-standard growth. Preprint (2013)Google Scholar
  24. 24.
    Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710, 877, 1986. English translation: Math. USSR-Izv. 29(3), 3–66 (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of JyvaskylaJyvaskylaFinland

Personalised recommendations