Lower semicontinuity via W1,q-quasiconvexity

We isolate a general condition, that we call “localization principle”, on the integrand L:M→[0,∞], assumed to be continuous, under which W1,q-quasiconvexity with q∈[1,∞] is a sufficient condition for to be sequentially weakly lower semicontinuous on W1,p(Ω;Rm) with p∈]1,∞[. Some applications are given.