Skip to main content
SpringerLink
Log in

On Hölder Continuity of Vector-Valued Minimizers for Quadratic Growth Functionals

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In dimension \(n\le 4\), we give a few sufficient conditions for interior everywhere Hölder continuity of weak vector-valued minimizers of a class of quadratic growth functionals with VMO dependence on variable x.

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. Campanato, S.: Sistemi ellittici in forma divergenza. Regolarità all’interno. Quaderni Scuola Norm. Sup. Pisa, Pisa (1980)

    MATH  Google Scholar 

  2. Campanato, S.: A maximum principle for non-linear elliptic systems: boundary fundamental estimates. Adv. Math. 66, 291–317 (1987)

    Article  MathSciNet  Google Scholar 

  3. Campanato, S.: Elliptic systems with nonlinearity \(q\) greater or equal to two. Regularity of the solution of the Dirichlet problem. Ann. Mat. Pura Appl. 147, 117–150 (1987)

  4. Daněček, J., Viszus, E.: \(C^{0,\gamma }\)—regularity for vector-valued minimizers of quasilinear functionals. NoDEA Nonlinear Differ. Equ. Appl. 16, 189–211 (2009)

    Article  Google Scholar 

  5. Daněček, J., Viszus, E.: On Hölder regularity for vector-valued minimizers of quasilinear functionals. Math. Bohemica 135(2), 199–207 (2010)

    Article  MathSciNet  Google Scholar 

  6. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. In: Annals of Mathematics Studied, vol. 105. Princeton University Press, Princeton (1983)

  7. Giaquinta, M., Giusti, E.: Differentiability of minima of non-differentiable functionals. Invent. Math. 72, 285–298 (1983)

    Article  MathSciNet  Google Scholar 

  8. Giusti, E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Officine Grafiche Tecnoprint Bologna (1994)

  9. Giusti, E., Miranda, M.: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale di calcolo delle variazioni ellittico. Boll. Unione Mat. Ital. 12, 219–226 (1968)

    MATH  Google Scholar 

  10. Goodrich, C., Scilla, G., Stroffolini, B.: Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth. Proc. R. Soc. Edinb. A: Math. (2021). https://doi.org/10.1017/prm.2021.53

  11. Jost, J., Meier, M.: Boundary regularity for minima of certain quadratic functionals. Math. Ann. 262, 549–561 (1983)

    Article  MathSciNet  Google Scholar 

  12. Kristensen, J., Melcher, C.: Regularity in oscillatory nonlinear elliptic systems. Math. Zeitsch. 260(4), 813–847 (2008)

    Article  MathSciNet  Google Scholar 

  13. Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)

    Article  MathSciNet  Google Scholar 

  14. Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Prague (1977)

    MATH  Google Scholar 

  15. Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–425 (2006)

    Article  MathSciNet  Google Scholar 

  16. Nečas, J., Stará, J.: Principio di massimo per i sistemi ellittici quasilineari non diagonali. Boll. Unione Mat. Ital. (4) 6, 1–10 (1972)

    MATH  Google Scholar 

  17. Ragusa, M.A., Tachikawa, A.: On continuity of minimizers for certain quadratic growth functionals. J. Math. Soc. Jpn. (3) 57, 691–700 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Ragusa, M.A., Tachikawa, A.: Regularity of the minimizers of some variational integrals with discontinuity. Z. Anal. Anwend. 27, 469–482 (2008)

    Article  MathSciNet  Google Scholar 

  19. Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)

    Article  MathSciNet  Google Scholar 

  20. Šverák, V., Yan, X.: Non-Lipschitz minimizers of smooth strongly convex functionals. Proc. Natl. Acad. Sci. USA 99(24), 15269–15276 (2002)

    Article  MathSciNet  Google Scholar 

  21. Ziemer, W.P.: Weakly Differentiable Functions. Springer, Heidelberg (1989)

    Book  Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous referee for valuable suggestions and comments which contribute to improve the readability of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics Comenius University, Mlynská Dolina, 84248, Bratislava, Slovak Republic

    Eugen Viszus

Authors
  1. Eugen Viszus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eugen Viszus.

Additional information

Publisher's Note

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

The author was supported by the research project Slovak Grant Agency No. 1/0358/20.

Rights and permissions

Springer Nature or its licensor 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

Viszus, E. On Hölder Continuity of Vector-Valued Minimizers for Quadratic Growth Functionals. Mediterr. J. Math. 19, 235 (2022). https://doi.org/10.1007/s00009-022-02126-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-022-02126-y

Keywords

Mathematics Subject Classification

Search

Navigation