### Abstract

We consider non-autonomous functionals F (u; Omega) = integral(Omega) f(x, Du) dx, where the density f : Omega x R(nN) -> R has almost linear growth, i.e., f(x, xi) approximate to vertical bar xi vertical bar log (1 + vertical bar xi vertical bar). We prove partial C (1,gamma) -regularity for minimizers u: R(n) superset of Omega -> R(N) under the assumption that D (xi) f (x, xi) is Holder 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.

Original language | English |
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Pages (from-to) | 83-114 |

Number of pages | 32 |

Journal | Manuscripta Mathematica |

Volume | 136 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Sep 2011 |

### Keywords

- VARIATIONAL INTEGRALS
- CASE 1-LESS-THAN-P-LESS-THAN-2
- CONVEX INTEGRALS
- SINGULAR SET
- MINIMIZERS

## Cite this

Breit, D., De Maria, B., & di Napoli, A. P. (2011). Regularity for non-autonomous functionals with almost linear growth.

*Manuscripta Mathematica*,*136*(1-2), 83-114. https://doi.org/10.1007/s00229-011-0432-2