Lower semicontinuity via W^{1,q}-quasiconvexity

We isolate a general condition, that we call "localization principle", on the integrand $L:\mathbb{M}\to[0,\infty]$, 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 wea...

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Published in:Bulletin des sciences mathématiques 2013-07, Vol.137 (5), p.602-616
Main Author: Mandallena, Jean-Philippe
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description We isolate a general condition, that we call "localization principle", on the integrand $L:\mathbb{M}\to[0,\infty]$, 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 $W^{1,p}(\Omega;\mathbb{R}^m)$ with $p\in]1,\infty[$. Some applications are given.
doi_str_mv 10.1016/j.bulsci.2012.12.004
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title Lower semicontinuity via W^{1,q}-quasiconvexity
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