manuscripta mathematica

, Volume 126, Issue 1, pp 1–40

Partial regularity for minima of higher order functionals with p(x)-growth

Article

Abstract

For higher order functionals \(\int_\Omega f(x, \delta u(x), {D^m}u(x))\,dx\) with p(x)-growth with respect to the variable containing Dmu, we prove that Dmu is Hölder continuous on an open subset \(\Omega_0 \subset \Omega\) of full Lebesgue-measure, provided that the exponent function \(p : \Omega \to (1, \infty)\) itself is Hölder continuous.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E. and Fusco N. (1987). A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal. 99: 261–28 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Acerbi E. and Mingione G. (2001). Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156: 121–140 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Acerbi E. and Mingione G. (2001). Regularity results for a class of quasiconvex functionals with nonstandard growth. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30(4): 311–339 MATHMathSciNetGoogle Scholar
  4. 4.
    Acerbi E. and Mingione G. (2002). Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164: 213–259 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Campanato S. (1965). Equazioni ellittiche del II ordine e spazi \({\mathcal{L}}^{(2,\lambda)}\). Ann. Mat. Pura Appl. 69: 321–382 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Carozza M., Fusco N. and Mingione G. (1998). Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Mat. Pura Appl. 175: 141–164 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coscia A. and Mingione G. (1999). Hölder continuity of the gradient pf p(x)-harmonic mappings. C. R. Acad. Sci. Paris Ser. I Math. 328: 363–368 MATHMathSciNetGoogle Scholar
  8. 8.
    Duzaar F., Grotowski J.F. and Kronz M. (2004). Partial and full boundary regularity for minimizers of functionals with nonquadratic growth. J. Convex Anal. 11: 1–40 MathSciNetGoogle Scholar
  9. 9.
    Duzaar, F., Grotowski, J.F., Kronz, M.: Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Ann. Mat. Pura Appl. IV. Ser., 184(4), (2005).Google Scholar
  10. 10.
    Ekeland I. (1979). Nonconvex minimization problems. Bill. Am. Math. Soc. 1: 443–474 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Eleuteri, M.: Hölder continuity results for a class of functionals with nonstandard growth. Boll. Unione Mat. Ital. 8(7-B), (2004)Google Scholar
  12. 12.
    Evans, L.C., Gariepy, R.F.: Blowup, compactness and partial regularity in the calculus of variations. Indiana Univ. Math. J. 36(2) (1987)Google Scholar
  13. 13.
    Fusco N. and Hutchinson J. (1985). C 1,α partial regularity of functions minimizing quasiconvex integrals. Manuscr. Math. 54: 121–143 CrossRefMathSciNetGoogle Scholar
  14. 14.
    Giaquinta M. (1983). Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton MATHGoogle Scholar
  15. 15.
    Giaquinta M. and Modica G. (1979). Regularity results for some classes of higher order non linear elliptic systems. J. Reine Angew. Math. 311–312: 145–169 MathSciNetGoogle Scholar
  16. 16.
    Habermann J.: Regularity results for functionals and Calderón–Zygmund estimates for systems of higher order with p(x)-growth. Dissertation, University of Erlangen (2006)Google Scholar
  17. 17.
    Kronz, M.: Partial regularity results for minimizers of quasiconvex functionals of higher order. Ann. I. H. Poincaré, 19(1) (2002)Google Scholar
  18. 18.
    Kronz, M.: Quasimonotone Systems of Higher Order. Boll. UMI 8(6-B) (2003)Google Scholar
  19. 19.
    Kronz, M.: Habilitationsschrift, Universität Erlangen (to appear)Google Scholar
  20. 20.
    Marcellini P. (1989). Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Ration. Mech. Anal. 105: 267–284 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rajagopal K.R. and Ru̇žička M. (2001). Mathematical modelling of electro-rheological fluids. Cont. Mech. Therm. 13: 59–78 MATHCrossRefGoogle Scholar
  22. 22.
    Ru̇žička M. (2000). Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Heidelberg Google Scholar
  23. 23.
    Ziemer W. (1989). Weakly Differentiable Functions. Springer, Heidelberg MATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute for MathematicsFriedrich-Alexander UniversityErlangenGermany

Personalised recommendations