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 supA→∞ F(A)/|A|p = 0 (*) for some p ∈ ] 1, ∞ [. For an open and bounded set Ω ⊂ ℝm, we show that if q≧ m-1/m p and q > 1, then the variational integral ℱ(u; Ω):= ∫Ω F(Du) dx is lower semicontinuous on sequences of W1,p functions converging weakly in W1,q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Malỳ [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