Advertisement

Manuscripta Mathematica

, Volume 136, Issue 1–2, pp 83–114 | Cite as

Regularity for non-autonomous functionals with almost linear growth

  • Dominic BreitEmail author
  • Bruno De Maria
  • Antonia Passarelli di Napoli
Article

Abstract

We consider non-autonomous functionals \({\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}\), where the density \({f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}\) has almost linear growth, i.e.,
$$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$
We prove partial C 1,γ -regularity for minimizers \({u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}\) under the assumption that D ξ f (x, ξ) is Hölder continuous with respect to the x-variable. If the x-dependence is C 1 we can improve this to full regularity provided additional structure conditions are satisfied.

Mathematics Subject Classification (2000)

35B65 35J50 49J25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E., Fusco N.: Regularity for minimizers of non-quadratic functionals. The case 1 < p < 2. J. Math. Anal. Appl. 1, 115–135 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Acerbi E., Fusco N.: Partial regularity under anisotropic (p,q)-growth conditions. J. Differ. Equ. 107, 46–67 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  4. 4.
    Bildhauer, M.: Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions. Lecture Notes in Mathematics 1818. Springer-Verlag, Berlin (2002)Google Scholar
  5. 5.
    Bildhauer M., Fuchs M.: Partial regularity for variational integrals with (s, μ, q)-growth. Calc. Var. Partial Differ. Equ. 13, 537–560 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bildhauer M., Fuchs M.: C 1,α-solutions to non-autonomous anisotropic variational problems. Calc. Var. Partial Differ. Equ. 24(3), 309–340 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Breit, D.: New regularity theorems for non-autonomous anisotropic variational integrals. Preprint 241, Saarland University (2009)Google Scholar
  8. 8.
    Carozza M., Passarelli di Napoli A.: A regularity theorem for minimizers of quasiconvex integrals: the case 1 < p < 2. Proc. R. Soc. Edinb. 126, 1181–1199 (1996)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carozza M., Fusco N., Mingione G.: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Mat. Pura Appl. CLXXV, 141–164 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. It. 1(4), 135–137 (1968)MathSciNetzbMATHGoogle Scholar
  11. 11.
    De Maria, B.: A regularity result for a convex functional and bounds for the singular set. ESAIM: COCV (2009). doi: 10.1051/cocv/030, 2009
  12. 12.
    De Maria, B., Passarelli di Napoli, A.: Partial regularity for non autonomous functionals with non standard growth conditions. Calc. Var. Partial Differ. Equ. (2009). doi: 10.1007/s00526-009-0293-7
  13. 13.
    De Maria, B., Passarelli di Napoli, A.: A new partial regularity for non autonomous functionals with non standard growth conditions. Preprint CVGMT (2010)Google Scholar
  14. 14.
    Duzaar F., Grotowski J.F., Kronz M.: Regularity for almost minimizers of quasi-convex variational integrals with subquadratic growth. Annali di Matematica 184, 421–448 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Esposito L., Mingione G.: Partial regularity for minimizers of convex integrals with L log L-growth. Nonlinear Differ. Equ. Appl. 7, 107–125 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Esposito L., Leonetti F., Mingione G.: Regularity results for minimizers of irregular integral functionals with (p,q) growth. Forum Math. 14, 245–272 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Esposito L., Leonetti F., Mingione G.: Sharp regularity for functionals with (p,q) growth. J. Differ. Equ. 204, 5–55 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fuchs M., Mingione G.: Full C 1, α-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth. Manuscripta Math. 102, 227–250 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Fuchs M., Osmolovsky V.: Variational integrals on Orlicz-Sobolev spaces. Z. Anal. Anwend. 17, 393–415 (1998)zbMATHGoogle Scholar
  20. 20.
    Fuchs M., Seregin G.: A regularity theory for variational integrals with L log L-growth. Calc. Var. Partial Differ. Equ. 6, 171–187 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Giusti E.: Direct Methods in the Calculus of Variations. World Scientific, River Edge (2003)zbMATHCrossRefGoogle Scholar
  22. 22.
    Greco L., Iwaniec T., Sbordone C.: Variational integrals of nearly linear growth. Differ. Integral Equ. 10(4), 687–716 (1997)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kristensen J., Mingione G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Marcellini P.: Regularity and existence of solutions of elliptic equations with (p,q)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mingione G.: The singular set of solutions to non differentiable elliptic systems. Arch. Ration. Mech. Anal. 166, 287–301 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mingione G.: Regularity of minima: an invitation to the dark side of calculus of variations. Appl. Math. 51, 355–426 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mingione G., Siepe F.: Full C 1, α-regularity for minimizers of integral functionals with L log L-growth. Z. Anal. Anwed. 18(4), 1083–1100 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Passarelli di Napoli A., Siepe F.: A regularity result for a class or anisotropic systems. Rend. Ist. Mat. di Trieste 28, 13–31 (1997)MathSciNetGoogle Scholar
  29. 29.
    Schmidt T.: Regularity of minimizers of W 1,p-quasiconvex variational integrals with (p, q)- growth. Calc. Var. Partial Differ. Equ. 1, 1–24 (2008)CrossRefGoogle Scholar
  30. 30.
    Stampacchia G.: Le problème de Dirichlet pour les equations elliptiques du second ordre à coefficients dicontinus. Ann Inst. Fourier Grenoble 15(1), 189–258 (1965)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Dominic Breit
    • 1
    Email author
  • Bruno De Maria
    • 2
  • Antonia Passarelli di Napoli
    • 2
  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università di Napoli “Federico II”NaplesItaly

Personalised recommendations