Skip to main content
SpringerLink
Log in

C1,α-solutions to non-autonomous anisotropic variational problems

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

Abstract

We establish several smoothness results for local minimizers of non-autonomous variational integrals with anisotropic growth conditions.

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., Fusco, N.: Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140, 115–135 (1989)

    Article  Google Scholar 

  2. Bildhauer, M.: Convex variational problems: linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, Berlin Heidelberg New York (2003)

  3. Bildhauer, M., Fuchs, M.: Partial regularity for variational integrals with (s,μ,q)-growth. Calc. Var. 13, 537–560 (2001)

    Article  Google Scholar 

  4. Bildhauer, M., Fuchs, M.: Twodimensional anisotropic variational problems. Calc. Var. 16, 177–186 (2003)

    Article  Google Scholar 

  5. Bildhauer, M., Fuchs, M.: A regularity result for stationary electrorheological fluids in two dimensions. Math. Meth. Appl. Sci. 27, 1607–1617 (2004)

    Article  Google Scholar 

  6. Bildhauer, M., Fuchs, M.: Lavrentiev phenomenon, relaxation and some regularity results for anisotropic functional. Preprint 103, Saarland University (see http://www.math. uni-sb.de/preprint.html)

  7. Bildhauer, M., Fuchs, M., Mingione, G.: A priori gradient bounds and local C1,α-estimates for (double) obstacle problems under nonstandard growth conditions. Z. Anal. Anw. 20(4), 959–985 (2001)

    Google Scholar 

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

    Article  Google Scholar 

  9. Coscia, A., Mingione, M.: Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris Sr. I Math. 328, 363–368 (1999)

    Google Scholar 

  10. Cupini, G., Guidorzi, M., Mascolo, E.: Regularity of minimizers of vectorial integrals with p-q growth. Nonlinear Anal. 54, 591–616 (2003)

    Article  Google Scholar 

  11. Dacorogna, B.: Direct methods in the calculus of variations. Applied Mathematical Sciences 78, Springer, Berlin Heidelberg New York (1989)

  12. Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity for functionals with (p, q) growth. J. Diff. Eq. 204, 5–55 (2004)

    Article  Google Scholar 

  13. Fonseca, I., Malý, J., Mingione, G.: Scalar minimizers with fractal singular sets. Arch. Rat. Mech. Anal. 172, 295–312 (2004)

    Article  Google Scholar 

  14. Frehse, J.: Two dimensional variational problems with thin obstacles. Math. Z. 143, 279–288 (1975)

    Article  Google Scholar 

  15. Frehse, J., Seregin, G.: Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening. Proc. St. Petersburg Math. Soc. 5, 184–222 (1998) (in Russian). English translation: Transl. Am Math. Soc., II 193, 127–152 (1999)

    Google Scholar 

  16. Giaquinta, M.: Multiple Integrals in The Calculus of Variations and Nonlinear Elliptic Systems. Ann. Math. Studies 105, Princeton University Press, Princeton (1983)

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

    Article  Google Scholar 

  18. Hamburger, C.: Regularity of differential forms minimizing degenerate elliptic functionals. J. reine angew. Math. 431, 7–64 (1992)

    Google Scholar 

  19. Ladyzhenskaya, O.A., Ural'tseva, N.N.: Linear and Quasilinear Elliptic Equations. Nauka, Moskow, 1964. English translation: Academic Press, New York (1968)

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

    Article  Google Scholar 

  21. Marcellini, Paolo: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23(4), 1–25 (1996)

    Google Scholar 

  22. Mingione, G., Siepe, F.: Full C1,α-regularity for minimizers of integral functionals with L log L growth. Z. Anal. Anw. 18, 1083–1100 (1999)

    Google Scholar 

  23. Seregin, G.A.: Differential properties of solutions of variational problems for functionals with linear growth. Problemy Matematicheskogo Analiza, Vypusk 11, Isazadel'stvo LGU (1990), 51–79 (in Russian). English translation: J. Soviet Math. 64, 1256–1277 (1993)

    Article  Google Scholar 

  24. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus. Ann. Inst. Fourier Grenoble 15.1, 189–258 (1965)

    Google Scholar 

  25. Tolksdorf, P.: Everywhere-regularity for some quasilinear systems with a lack of ellipticity. Ann. Mat. Pura Appl. 134, 241–266 (1983)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saarland University, P.O. Box 151150, 66041, Saarbrücken, Germany

    Michael Bildhauer & Martin Fuchs

Authors
  1. Michael Bildhauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Fuchs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Bildhauer.

Additional information

Mathematics Subject Classification (2000) 49N60, 49N99

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bildhauer, M., Fuchs, M. C1,α-solutions to non-autonomous anisotropic variational problems. Calc. Var. 24, 309–340 (2005). https://doi.org/10.1007/s00526-005-0327-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-005-0327-8

Keywords

Search

Navigation