### Abstract

Let there be given a non-negative, quasiconvex function F satisfying the growth condition lim sup_{A?8} F(A)/|A|^{p} = 0 (*) for some p ? ] 1, 8 [. For an open and bounded set O ? R^{m}, we show that if q? m-1/m p and q > 1, then the variational integral F(u; O):= ?_{O} F(Du) dx is lower semicontinuous on sequences of W^{1,p} functions converging weakly in W^{1,q}. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Mal? [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.

Original language | English |
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Pages (from-to) | 797-817 |

Number of pages | 21 |

Journal | Proceedings of the Royal Society of Edinburgh, Section A: Mathematics |

Volume | 127 |

Issue number | 4 |

Publication status | Published - 1997 |

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## Cite this

*Proceedings of the Royal Society of Edinburgh, Section A: Mathematics*,

*127*(4), 797-817.