Skip to main content
SpringerLink
Log in

Weighted Lorentz estimates for non-uniformly elliptic problems with variable exponents

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper, a global \(L^{s,t}_\omega \)-bound for the gradient of weak solutions to non-uniformly nonlinear elliptic equations with variable exponents is presented. The main difficulties arise from variable \((p(\cdot ),q(\cdot ))\)-growth conditions can be handled by standard techniques. Under the appropriated assumptions and minimal regularity on initial data of the problem, weighted regularity estimates in the frame of Lorentz spaces will be established. Furthermore, the use of level-set inequalities on distribution functions is also imposed to obtained the norm bounds in a wide range of generalized Lebesgue spaces such as Lorentz, Lorentz-Morrey or Orlicz spaces, etc.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ball, J.M.: Some open problems in elasticity. In: Newton, P. (ed.) Geometry, Mechanics, and Dynamics, pp. 3–59. Springer-Verlag, New York (2002)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. Partial Differ. Equ. 57, 57–62 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beck, L., Mingione, G.: Lipschitz bounds and non-uniform ellipticity. Comm. Pure Appl. Math. 73, 944–1034 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byun, S.-S., Lee, H.-S.: Calderón-Zygmund estimates for elliptic double phase problems with variable exponents. J. Math. Anal. Appl. 501(1), 124015 (2021)

    Article  MATH  Google Scholar 

  10. Byun, S.-S., Lee, H.-S.: Gradient estimates of \(\omega \)-minimizers to double phase problems with variable exponents. Quart. J. Math. 72(4), 1191221 (2021)

    Article  MathSciNet  Google Scholar 

  11. Byun, S.-S., Oh, J.: Global gradient estimates for non-uniformly elliptic equations. Calc. Var. Partial Differ. Equ. 56, 46 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations. Am. Math. Soc. 43(1), 1–21 (1995)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Crespo-Blanco, Á., Gasiński, L., Harjulehto, P., Winkert, P.: A new class of double phase variable exponent problems: existence and uniqueness, preprint, (2021), arXiv:2103.08928

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. De Filippis, C., Mingione, G.: A borderline case of Calderón-Zygmund estimates for non-uniformly elliptic problems. St. Petersburg Math. J. 31(3), 82–115 (2019)

    Google Scholar 

  19. Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  20. Fang, Y., Radulescu, V., Zhang, C.: Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent, Preprint. (2021)

  21. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson/Prentice Hall, New Jersey (2004)

    MATH  Google Scholar 

  22. Harjulehto, P., Hästö, P.: Orlicz Spaces and Generalized Orlicz Spaces, vol. 2236. Springer, Cham (2019)

    MATH  Google Scholar 

  23. Kuusi, T., Mingione, G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4(1), 1–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, W., Dai, G.: Existence and multiplicity results for double phase problem. J. Differ. Equ. 265(9), 4311–4334 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s inequality for double phase functionals with variable exponents. Forum Math. 31(2), 517–527 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Marcellini, P.: Regularity of minimisers of integrals of the calculus of variations with non-standard growth conditions. Arch. Rat. Mech. Anal. 105, 267–284 (1989)

    Article  MATH  Google Scholar 

  27. Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mingione, G., Rădulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501(1), 125197 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Muckenhoupt, B., Wheeden, R.L.: Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192, 261–275 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nguyen, T.-N., Tran, M.-P.: Level-set inequalities on fractional maximal distribution functions and applications to regularity theory. J. Funct. Anal. 280(1), 108797 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rădulescu, V.D., Repovs, D.D.: Partial differential equations with variable exponents. Variational methods and qualitative analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, Fla., (2015). https://doi.org/10.1201/b18601

  32. Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9(1), 710–728 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ruzicka, M.: Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, vol. 1748. Springer-Verlag, Berlin (2000)

  34. Tachikawa, A.: Boundary regularity of minimizers of double phase functionals. J. Math. Anal. Appl. 501(1), 123946 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tran, M.-P., Nguyen, T.-N.: Global Lorentz estimates for non-uniformly nonlinear elliptic equations via fractional maximal operators. J. Math. Anal. Appl. 501(1), 124084 (2021)

    Article  MATH  Google Scholar 

  36. Tran, M.-P., Nguyen, T.-N.: Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data. Nonlinear Anal. 214(3), 112613 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tran, M.-P., Nguyen, T.-N.: Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces. J. Math. Anal. Appl. 509(1), 125928 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tran, M.-P., Nguyen, T.-N., Huynh, P.-N.: Calderón-Zygmund-type estimates for singular quasilinear elliptic obstacle problems with measure data, arXiv:2109.01026

  39. Tran, M.-P., Nguyen, T.-N., Nguyen, G.-B.: Lorentz gradient estimates for a class of elliptic \(p\)-Laplacian equations with a Schrödinger term. J. Math. Anal. Appl. 496(1), 124806 (2021)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  41. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710 (1986)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is funded by Ho Chi Minh City University of Education Foundation for Science and Technology under grant number CS.2021.19.02TD.

Author information

Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Minh-Phuong Tran

  2. Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    Thanh-Nhan Nguyen

  3. Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    Le-Tuyet-Nhi Pham & Thi-Thanh-Truc Dang

Authors
  1. Minh-Phuong Tran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thanh-Nhan Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Le-Tuyet-Nhi Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thi-Thanh-Truc Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thanh-Nhan Nguyen.

Ethics declarations

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tran, MP., Nguyen, TN., Pham, LTN. et al. Weighted Lorentz estimates for non-uniformly elliptic problems with variable exponents. manuscripta math. 172, 1227–1244 (2023). https://doi.org/10.1007/s00229-022-01452-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-022-01452-5

Mathematics Subject Classification

Search

Navigation