  We isolate a general condition, that we call "localization principle", on the
integrand L:\MM\to[0,\infty], assumed to be continuous, under which
W^{1,q}-quasiconvexity with q\in[1,\infty] is a sufficient condition for
I(u)=\int_\Omega L(\nabla u(x))dx to be sequentially weakly lower
semicontinuous on W^{1,p}(\Omega;\RR^m) with p\in]1,\infty[. Some applications
are given.
