Skip to main content

Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions

Inspired by the work of Marcellini and Papi, we consider local minima u: ℝn ⊃ Ω → ℝM of variational integrals \( \int\limits_\Omega {h\left( {\left| {\nabla u} \right|} \right)} \) dx and prove interior gradient bounds under rather general assumptions on h provided that u is locally bounded. Our requirements on the density h do not involve the dimension n. Bibliography: 18 titles.

This is a preview of subscription content, access via your institution.


  1. 1.

    M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton (1983).

    MATH  Google Scholar 

  2. 2.

    E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, New Jersey (2003).

    MATH  Google Scholar 

  3. 3.

    K. Uhlenbeck, “Regularity for a class of nonlinear elliptic systems,” Acta Math. 138, 219–240 (1977).

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    M. Giaquinta, G. Modica, “Remarks on the regularity of the minimizers of certain degenerate functionals,” Manus. Math. 57, 55–99 (1986).

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    P. Marcellini, “Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions,” Arch. Rat. Mech. Anal. 105, 267–284 (1989).

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    P. Marcellini, “Regularity for elliptic equations with general growth conditions,” J. Differ. Equ. 105, 296–333 (1993).

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    P. Marcellini, “Everywhere regularity for a class of elliptic systems without growth conditions,” Ann. Scuola Norm. Sup. Pisa 23, 1–25 (1996).

    MATH  MathSciNet  Google Scholar 

  8. 8.

    G. Mingione, F. Siepe, “Full C 1 regularity for minimizers of integral functionals with Llog L growth.” Z. Anal. Anw. 18, 1083–1100 (1999).

    MATH  MathSciNet  Google Scholar 

  9. 9.

    P. Marcellini, G. Papi, “Nonlinear elliptic systems with general growth,” J. Differ. Equ. 221, 412–443 (2006).

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    R. A. Adams, Sobolev Spaces, Academic Press, New York etc. (1975).

    MATH  Google Scholar 

  11. 11.

    A. Dall’Aglio, E. Mascolo, G. Papi, “Local boundedness for minima of functionals with nonstandard growth conditions,” Rend. Mat. 18, 305–326 (1998).

    MATH  MathSciNet  Google Scholar 

  12. 12.

    M. Bildhauer, Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions, Lect. Notes Math. 1818, Springer, Berlin etc. (2003).

  13. 13.

    M. Bildhauer, M. Fuchs, “Elliptic variational problems with nonstandard growth” In: Nonlinear Problems in Mathematical Physics and Related Topics I. In Honor of Prof. O.A. Ladyzhenskaya. Int. Math. Ser. 1, pp. 49–62. Springer, New York (2002).

  14. 14.

    A. D’Ottavio, F. Leonetti, C. Musciano, “Maximum principle for vector valued mappings minimizing variational integrals,” Atti Semin. Mat. Fis. Univ. Modena 46, 677–683 (1998).

    MATH  MathSciNet  Google Scholar 

  15. 15.

    M. Fusch, “A note on non-uniformly elliptic Stokes-type systems in two variables,” J. Math. Fluid Mech. [To appear]

  16. 16.

    H. J. Choe, “Interior behaviour of minimizers for certain functionals with nonstandard growth,” Nonlinear Anal., Theory Methods Appl. 19, No. 10, 933–945 (1992).

    MATH  Article  MathSciNet  Google Scholar 

  17. 17.

    S. Campanato, “Hölder continuity of the solutions of some non-linear elliptic systems,” Adv. Math. 48, 16–43 (1983).

    Article  MathSciNet  Google Scholar 

  18. 18.

    M. Bildhauer, M. Fuchs, G. Mingione, “Apriori gradient bounds and local C 1-estimates for (double) obstacle problems under nonstandard growth conditions,” Z. Anal. Anw. 20, No. 4, 959–985 (2001).

    MATH  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to D. Apushkinskaya.

Additional information

Translated from Problems in Mathematical Analysis 43, November 2009, pp. 35–50.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Apushkinskaya, D., Bildhauer, M. & Fuchs, M. Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions. J Math Sci 164, 345–363 (2010).

Download citation


  • Strict Convexity
  • Young Inequality
  • Nonlinear Elliptic System
  • Local Boundedness
  • Nonstandard Growth