Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand

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.