%0 Journal Article %T Lower semicontinuity via W^{1,q}-quasiconvexity %A Jean-Philippe Mandallena %J Mathematics %D 2011 %I arXiv %X 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. %U http://arxiv.org/abs/1106.2828v7