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

Abstract

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, gL t(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.

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References

  1. 1.

    Aboulaïch, R., Meskine, D., Souissi, A.: New diffusion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Adamowicz, T., Toivanen, O.: Hölder continuity of quasiminimizers with nonstandard growth. Nonlinear Anal. 125, 433–456 (2015)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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). MR2790542

    Book  Google Scholar 

  11. 11.

    Fan, X.L., Zhao, D.: A class of de Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295–318 (1999)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Giusti, E.: Direct methods in the calculus of variations. World Scientific, Singapore (2003)

    Book  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)

    MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Rajagopal, K.R., Ru̇žčka, M.: On the modeling of electrorheological materials. Mech. Research Comm. 23, 401–407 (1996)

    Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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 

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Correspondence to Visa Latvala.

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Latvala, V., Toivanen, O. Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure. Potential Anal 47, 21–36 (2017). https://doi.org/10.1007/s11118-016-9606-6

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Keywords

  • 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