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Hölder regularity of quasiminimizers under generalized growth conditions

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Abstract

We prove Harnack’s inequality for local (quasi)minimizers in generalized Orlicz spaces without polynomial growth or coercivity conditions. As a consequence, we obtain the local Hölder continuity of local (quasi)minimizers. The results include as special cases standard, variable exponent and double phase growth.

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References

  1. Alaouia, M.K., Nabilab, T., Altanjia, M.: On some new non-linear diffusion models for the image filtering. Appl. Anal. 93(2), 269–280 (2014)

    Article  MathSciNet  Google Scholar 

  2. Avci, M., Pankov, A.: Multivalued elliptic operators with nonstandard growth. Adv Nonlinear Anal. (2016). doi:10.1515/anona-2016-0043

    Google Scholar 

  3. Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St Petersburg Math. J. 27, 347–379 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Björn, A., Björn, J.: Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich (2011)

  6. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215(2), 443–496 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218(1), 219–273 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colombo, M., Mingione, G.: Calderón-Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270, 1416–1478 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue spaces, Foundations and harmonic analysis. Birkhäuser/Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  11. Cruz-Uribe, D., Hästö, P.: Extrapolation and interpolation in generalized Orlicz spaces. Trans. Am. Math. Soc. (to appear)

  12. Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)

  13. Fan, X.-L.: An imbedding theorem for Musielak–Sobolev spaces. Nonlinear Anal. 75(4), 1959–1971 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giannetti, F., Passarelli di Napoli, A.: Regularity results for a new class of functionals with non-standard growth conditions. J. Differ. Eq. 254, 1280–1305 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  16. Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  17. Gwiazda, P., Wittbold, P., Wróblewska, A., Zimmermann, A.: Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. J. Differ. Eq. 253(2), 635–666 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gwiazda, P., Wittbold, P., Wróblewska, A., Zimmermann, A.: Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. J. Differ. Eq. 253(9), 2734–2738 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gwiazda, P., Wittbold, P., Wróblewska, A., Zimmermann, A.: Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces. Nonlinear Anal. 129, 1–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Harjulehto, P., Hästö, P.: Riesz potential in generalized Orlicz Spaces. Forum Math. 29(1), 229–244 (2017)

    Article  MathSciNet  Google Scholar 

  21. Harjulehto, P., Hästö, P., Klén, R.: Generalized Orlicz spaces and related PDE. Nonlinear Anal. 143, 155–173 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Harjulehto, P., Hästö, P., Latvala, V., Toivanen, O.: Critical variable exponent functionals in image restoration. Appl. Math. Lett. 26, 56–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hedberg, L.I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hästö, P.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal 269(12), 4038–4048 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hästö, P.: Corrigendum to “The maximal operator on generalized Orlicz spaces” [J. Funct. Anal. 269 (2015) 4038–4048]. J. Funct. Anal. 271(1), 240–243 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Krylov, N.V., Safonov, M.V.: A property of the solutions of parabolic equations with measurable coefficients, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44, 161–175 (1980)

    MathSciNet  Google Scholar 

  28. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces. Bull. Sci. Math. 137, 76–96 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Approximate identities and Young type inequalities in Musielak–Orlicz spaces. Czechoslovak Math. J. 63(138)(4), 933–948 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev and Trudinger type inequalities on grand Musielak–Orlicz–Morrey spaces. Ann. Acad. Sci. Fenn. Math. 40(1), 403–426 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Maeda, F.-Y., Ohno, T., Shimomura, T.: Boundedness of the maximal operator on Musielak–Orlicz–Morrey spaces. Tohoku Math. J. (to appear)

  32. Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Rational Mech. Anal. 105(3), 267–284 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Marcellini, P.: Regularity and existance of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Eq. 50(1), 1–30 (1991)

    Article  MATH  Google Scholar 

  34. Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Musielak, J.: Orlicz spaces and modular spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)

  36. Ohno, T., Shimomura, T.: Trudinger’s inequality for Riesz potentials of functions in Musielak–Orlicz spaces. Bull. Sci. Math. 138(2), 225–235 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ohno, T., Shimomura, T.: Musielak–Orlicz–Sobolev spaces on metric measure spaces. Czechoslovak Math. J. 65(140)(2), 435–474 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ok, J.: Gradient estimates for elliptic equations with \(L^{p(\cdot )}\log L\) growth. Calc. Var. Partial Differ. Eq. 55(2), 1–30 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ok, J.: Regularity results for a class of obstacle problems with nonstandard growth. J. Math. Anal. Appl. 444(2), 957–979 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pick, L., Růžička, M.: An example of a space \(L^{p(x)}\) on which the Hardy–Littlewood maximal operator is not bounded. Expo. Math. 19(4), 369–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Růžička, M.: Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)

  42. Świerczewska-Gwiazda, A.: Nonlinear parabolic problems in Musielak–Orlicz spaces. Nonlinear Anal. 98, 48–65 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yang, D., Yuan, W., Zhuo, C.: Musielak–Orlicz Besov-type and Triebel–Lizorkin-type spaces. Rev. Mat. Complut. 27(1), 93–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)

    MathSciNet  Google Scholar 

  45. Zhikov, V.V.: On Lavrentiev’s phenomenon. Russian J. Math. Phys. 3, 249–269 (1995)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Petteri Harjulehto & Peter Hästö

  2. Department of Mathematics, University of Oulu, 90014, Oulu, Finland

    Peter Hästö

  3. Institute of Mathematics, Polish Academy of Sciences, 00-956, Warsaw, Poland

    Olli Toivanen

Authors
  1. Petteri Harjulehto
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  2. Peter Hästö
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  3. Olli Toivanen
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Correspondence to Peter Hästö.

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Communicated by L. Ambrosio.

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Harjulehto, P., Hästö, P. & Toivanen, O. Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. 56, 22 (2017). https://doi.org/10.1007/s00526-017-1114-z

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