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

Abstract
Let there be given a non-negative, quasiconvex functionFsatisfying the growth condition for somep∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if then the variational integral is lower semicontinuous on sequences ofW1, pfunctions converging weakly inW1, q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Maly [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.

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