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


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 


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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

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