Skip to main content
SpringerLink
Log in

Weighted \(L^{p(\cdot )}\)-regularity for fully nonlinear parabolic equations

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove a global weighted \(L^{p(\cdot )}\)-regularity for the Hessian of strong solution to the Cauchy–Dirichlet problem for fully nonlinear parabolic equations in a bounded \(C^{1,1}\)-domain, where the associated nonlinearity is \((\delta ,R)\)-vanishing in independent variables, the variable exponent \(p(\cdot )\) is \(\log \)-Hölder continuous, and the weight \(\omega \) is of the \(A_{p(\cdot )/(n+1)}\) class. As a consequence, we also derive Morrey’s regularity for the Hessian of strong solution to this problem under consideration, which implies a global Hölder continuity of the spatial gradient under the assumption of higher regular datum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    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)

    MathSciNet  MATH  Google Scholar 

  3. Antontsev, S.N., Shmarev, S.I.: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60, 515–545 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Baroni, P.: Lorentz estimates for degenerate and singular evolutionary systems. J. Differ. Equ. 255, 2927–2951 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Bramanti, M., Cerutti, M.C.: \(W^{1,2}_{p}\)-solvability for the Cauchy–Dirichlet problem for parabolic equations with VMO coefficients. Commun. Partial Differ. Equ. 18, 1735–1763 (1993)

    MATH  Google Scholar 

  7. Byun, S.S., Lee, M.: Weighted estimates for nondivergence parabolic equations in Orlicz spaces. J. Funct. Anal. 269(8), 2530–2563 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Byun, S.S., Lee, M., Ok, J.: \(W^{2, p(\cdot )}\)-regulairty for elliptic equations in nondivergence form with BMO coefficients. Math. Ann. 363, 1023–1052 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Byun, S.S., Lee, M., Palagachev, D.K.: Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations. J. Differ. Equ. 260, 4550–4571 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Byun, S.S., Lee, M., Ok, J.: Nondivergence parabolic equations in weighted variable exponent spaces. Trans. Am. Math. Soc. 370, 2208–2293 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88(1), 85–139 (1952)

    MathSciNet  MATH  Google Scholar 

  12. Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. 130(2), 189–213 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Caffarelli, L.A.: Fully Nonlinear Elliptic Equations, vol. 43. American Mathematical Society Colloquium Publications (1996)

  14. Caffarelli, L.A., Huang, Q.: Estimates in the generalized Campamato–John–Nirenberg spaces for fully nonlinear elliptic equations. Duke Math. J. 118(1), 1–17 (2003)

    MathSciNet  MATH  Google Scholar 

  15. 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)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Coifman, R.R., Rochberg, R.: Another characterization of BMO. Proc. Am. Math. Soc. 79(2), 249–254 (1980)

    MathSciNet  MATH  Google Scholar 

  18. Crandall, M.G., Kocan, M., Świech, A.: \(L^{p}\)-theory for fully nonlinear uniformly parabolic equations. Commun. Partial Differ. Equ. 25(11–12), 1997–2053 (2000)

    MATH  Google Scholar 

  19. Diening, L., Lengeler, D., R\(\mathring{\text{u}}\)žička, M.: The Stokes and Poisson problem in variable exponent spaces. Complex Var. Elliptic Equ. 56(7–9), 789–811 (2011)

  20. Dong, H.: Solvability of second-order equations with hierarchically partially BMO coefficients. Trans. Am. Math. Soc. 364(1), 493–517 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Dong, H., Krylov, N.V.: On the existence of smooth solutions for fully nonlinear parabolic equations with measurable “coefficients” without convexity assumptions. Commun. Partial Differ. Equ. 38(6), 1038–1068 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Dong, H., Krylov, N.V.: Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces. Calc. Var. Partial Differ. Equ. 58(4), 145 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Dong, H., Krylov, N.V., Li, X.: On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains. St. Petersburg Math. J. 24(1), 39–69 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Escauriaza, L.: \(W^{2, n}\) a priori estimates for solutions to fully non-linear equations. Indiana Univ. Math. J. 42(2), 413–423 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Nauka, Moscow (1985). English transl., Math. Appl. (Soviet Ser.), vol. 7, D. Reidel Publ. Co., Dordrecht (1987)

  26. Krylov, N.V.: Parabolic and elliptic equations with VMO coefficients. Commun. Partial Differ. Equ. 32, 453–475 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Krylov, N.V.: On the existence of \(W^{2, p}\) solutions for fully nonlinear elliptic equations under relaxed convexity assumptions. Commun. Partial Differ. Equ. 38, 687–710 (2013)

    MATH  Google Scholar 

  28. Kim, D.: Parabolic equations with measurable coefficients II. J. Math. Anal. Appl. 334(1), 534–548 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Kim, D., Krylov, N.V.: Parabolic equations with measurable coefficients. Potential Anal. 26, 345–361 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Prasath, V.B.S., Vorotnikov, D.: On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: analysis. Nonlinear Anal. 168, 176–197 (2018)

    MathSciNet  MATH  Google Scholar 

  31. R\(\mathring{\text{ u }}\)žička, M.: Elektrorheological Fluids: Modeling and Mathematical Theory, Springer Lecture Notes in Math., vol. 1748. Springer, New York (2000)

  32. Wang, L.: On the regularity theory of fully nonlinear parabolic equations. Bull. Am. Math. Soc. 22(1), 107–114 (1990)

    MathSciNet  MATH  Google Scholar 

  33. Wang, L.: A geometric approach to the Calderón–Zygmund estimates. Acta Math. Sin. 19(2), 381–396 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Winter, N.: \(W^{2, p}\) and \(W^{1, p}\)-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations. Z. Anal. Anwend. 28(2), 129–164 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Zhikov, V.V.: Meyer-type estimates for solving the nonlinear Stokes system. Differ. Equ. 33(1), 108–115 (1997)

    MathSciNet  MATH  Google Scholar 

  36. Zhikov, V.V., Kozlov, S.M., Oleǐnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994)

    Google Scholar 

  37. Zhang, J., Zheng, S.: Lorentz estimates for fully nonlinear parabolic and elliptic equations. Nonlinear Anal. 148, 106–125 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Zhang, J., Zheng, S.: Lorentz estimates for asymptotically regularfully nonlinear parabolic equations. Math. Nachrichten 291(5–6), 996–1008 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for valuable comments on improving the quality of this paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050016, Hebei, China

    Junjie Zhang

  2. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Shenzhou Zheng

  3. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, 78539, USA

    Zhaosheng Feng

Authors
  1. Junjie Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shenzhou Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhaosheng Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaosheng Feng.

Additional information

Communicated by L. Caffarelli.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by NSF of China-10605218, 12001160 and 12071021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Zheng, S. & Feng, Z. Weighted \(L^{p(\cdot )}\)-regularity for fully nonlinear parabolic equations. Calc. Var. 59, 190 (2020). https://doi.org/10.1007/s00526-020-01848-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-020-01848-9

Mathematics Subject Classification

Search

Navigation