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Global weighted estimates for the nonlinear parabolic equations with non-standard growth

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Abstract

In this paper we consider a nonlinear parabolic equation of p(x, t)-Laplacian type in divergence form with measurable data over non-smooth domains. We establish the global Calderón–Zygmund theory for the weak solutions of such a problem in the setting of weighted Lebesgue spaces. The nonlinearity of the coefficients is assumed to be discontinuous with respect to (x, t)-variables and the lateral boundary of the domain is sufficiently flat beyond the Lipchitz category. As an application of the main result, the regularity in parabolic Morrey scales for the spatial gradient is also obtained.

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References

  1. Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136(2), 285–320 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baroni, P., Bögelein, V.: Calderón-Zygmund estimates for parabolic \(p(x, t)\)-Laplacian systems. Rev. Mat. Iberoam. 30(4), 1355–1386 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bögelein, V., Duzaar, F.: Highter integrability for parabolic systems with non-standard growth and degenerate diffusions. Publ. Mat. 55, 201–250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byun, S., Cho, Y., Palagachev, D.: Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains. Proc. Am. Math. Soc. 143(6), 2527–2541 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Byun, S., Ok, J., Ryu, S.: Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains. J. Differ. Equ. 254(11), 4290–4326 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Byun, S., Ok, J., Ryu, S.: Global gradient estimates for elliptic equations of \(p(x)\)-Laplacian type with BMO nonlinearity. J. Reine Angew. Math. 715, 1–38 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byun, S., Ok, J., Palagachev, D., Sofotova, L.: Parabolic systems with measurable coefficients in weighted Orlicz spaces. Commun. Contemp. Math. 18(2), 1550018 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byun, S., Ryu, S.: Global weighted estimates for the gradient of solutions to nonlinear elliptic equations. Ann. Inst. H. Poincare Anal. Non Lineaire 30(2), 291–313 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Byun, S., Wang, L.: Parabolic equations in Reifenberg domains. Arch. Ration. Mech. Anal. 176(2), 271–301 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Byun, S., Wang, L.: \(L^p\) estimates for parabolic equations in Reifenberg domains. J. Funct. Anal. 223(1), 44–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Byun, S., Wang, L.: Parabolic equations in time dependent Reifenberg domains. Adv. Math. 212(2), 797–818 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Byun, S., Wang, L.: Parabolic equations with BMO nonlinearity in Reifenberg domains. J. Reine Angew. Math. 615, 1–24 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51(1), 1–21 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. DiBenedetto, E., Friedman, A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357, 1–22 (1985)

    MathSciNet  MATH  Google Scholar 

  16. DiBenedetto, E., Manfredi, J.: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 155, 1107–1134 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

  18. Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong, G., Wang, L.: A geometric approach to the topological disk theorem for Reifenberg. Pac. J. Math. 233(2), 321–339 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Iwaniec, T.: Projections onto gradient fields and \(L^p\)-estimates for degenerated elliptic operators. Studia Math. 75, 293–312 (1983)

    MathSciNet  MATH  Google Scholar 

  21. Kinnunen, J., Zhou, S.: A local estimate for nonlinear equations with discontinuous coefficients. Commun. Partial Differ. Equ. 24(11–12), 2043–2068 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)

    MATH  Google Scholar 

  23. Lieberman, G.: Boundary and initial regularity for solutions of degenerate parabolic equations. Nonlinear Anal. 20(5), 551–569 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mengesha, T., Phuc, N.C.: Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains. J. Differ. Equ. 250(5), 2485–2507 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mengesha, T., Phuc, N.C.: Global estimates for quasilinear elliptic equations on Reifenberg flat domains. Arch. Ration. Mech. Anal. 203, 189–216 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Phuc, N.C.: Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10(1), 1–17 (2011)

  27. Phuc, N.C.: Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations. Adv. Math. 250, 387–419 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Misawa, M.: \(L^q\) estimates of gradients for evolutional \(p\)-Laplacian systems. J. Differ. Equ. 219, 390–420 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  30. Reifenberg, E.: Solution of the Plateau problem for \(m\)-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  31. Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)

  32. Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)

    Google Scholar 

  33. Torchinsky, A.: Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, vol. 123. Academic Press, Orlando (1986)

    MATH  Google Scholar 

  34. Toro, T.: Doubling and flatness: geometry of measures. Notices Am. Math. Soc. 44(9), 1087–1094 (1997)

    MathSciNet  MATH  Google Scholar 

  35. Yao, F.: Weighted gradient estimates for the parabolic \(p\)-Laplacian equations. Nonlinear Anal. 112, 58–68 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, C., Zhou, S.: Global weighted estimates for quasilinear elliptic equations with non-standard growth. J. Funct. Anal. 267, 605–642 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, C., Zhou, S., Xue, X.: Global gradient estimates for the parabolic \(p(x, t)\)-Laplacian equation. Nonlinear Anal. 105, 86–101 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author wishes to thank the anonymous reviewer for many valuable comments and suggestions to improve the expressions. This work was supported by the NSFC (No. 11671111), the Natural Science Foundation of Heilongjiang Province (QC2014C002), Heilongjiang Province Postdoctoral Special Science Foundation (LBHTZ0514) and PIRS of HIT (No. B201502).

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  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Chao Zhang

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  1. Chao Zhang
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Correspondence to Chao Zhang.

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Communicated by L. Caffarelli.

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Zhang, C. Global weighted estimates for the nonlinear parabolic equations with non-standard growth. Calc. Var. 55, 109 (2016). https://doi.org/10.1007/s00526-016-1066-8

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