Potential Analysis

, Volume 47, Issue 1, pp 21–36 | Cite as

Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure



We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of
$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$
with f subject to the general structural conditions
$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$
where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, gLt(Ω) for some t > n/p, and the lower weak Harnack estimate is proved under the stronger assumption that b, gL(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, gL(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.


Calculus of variations De Giorgi estimate Harnack inequality Lebesgue points Nonstandard growth p-Laplace p(⋅)-Laplacian Quasiminimizers Variable exponent Variational integral 

Mathematics Subject Classification (2010)

49N60 35J20 35J62 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aboulaïch, R., Meskine, D., Souissi, A.: New diffusion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Acerbi, E., Mingione, G., Seregin, G.A.: Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(1), 25–60 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Adamowicz, T., Toivanen, O.: Hölder continuity of quasiminimizers with nonstandard growth. Nonlinear Anal. 125, 433–456 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bollt, E., Chartrand, R., Esedoḡlu, S., Schultz, P., Vixie, K.: Graduated adaptive image denoising: Local compromise between total variation and isotropic diffusion. Adv. Comput. Math. 31(1–3), 61–85 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chiadò Piat, V., Coscia, A.: Hölder continuity of minimizers of functionals with variable growth exponent. Manuscripta Math. 93(3), 283–299 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    de Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. 3, 25–43 (1957)MathSciNetMATHGoogle Scholar
  9. 9.
    DiBenedetto, E., Trudinger, N.S.: Harnack inequalities for quasiminima of variational integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 295–308 (1984)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diening, L., Harjulehto, P., Hästö, P., Råžička, M.: Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics (2017). Springer-Verlag, Heidelberg (2011). MR2790542CrossRefGoogle Scholar
  11. 11.
    Fan, X.L., Zhao, D.: A class of de Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295–318 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fan, X.L., Zhao, D.: On the spaces L p(x)(Ω) and W m, p(x)(Ω). J. Math. Anal. Appl. 263, 424–446 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Giusti, E.: Direct methods in the calculus of variations. World Scientific, Singapore (2003)CrossRefMATHGoogle Scholar
  15. 15.
    Gong, J., Manfredi, J., Parviainen, M.: Nonhomogeneous variational problems and quasi-minimizers on metric spaces. Manuscripta Math. 137(1-2), 247–271 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Harjulehto, P., Hästö, P., Latvala, V.: Minimizers of the variable exponent, non-uniformly convex Dirichlet energy. J. Math. Pures Appl. 89(2), 174–197 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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(5), 731–742 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Harjulehto, P., Kinnunen, J., Lukkari, T.: Unbounded supersolutions of nonlinear equations with nonstandard growth. Bound. Value Probl. 48348, 20 (2007). doi:10.1155/2007/48348
  19. 19.
    Harjulehto, P., Kuusi, T., Lukkari, T., Marola, N., Parviainen, M.: Harnack’s inequality for quasiminimizers with non-standard growth conditions. J. Math. Anal. Appl. 344(1), 504–520 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)MATHGoogle Scholar
  21. 21.
    Kováčik, O., Rákosník, J.: On spaces L p(x) and W 1, p(x). Czechoslovak Math. J. 41(116, 4), 592–618 (1991)MathSciNetMATHGoogle Scholar
  22. 22.
    Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rajagopal, K.R., Ru̇žčka, M.: On the modeling of electrorheological materials. Mech. Research Comm. 23, 401–407 (1996)CrossRefMATHGoogle Scholar
  24. 24.
    Ru̇žčka, M.: Electrorheological fluids: modeling and mathematical theory, vol. 1748. Springer-Verlag, Berlin (2000)Google Scholar
  25. 25.
    Toivanen, O.: Local boundedness of general minimizers with nonstandard growth. Nonlinear Anal. 81, 62–69 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland
  2. 2.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland

Personalised recommendations