Lower semicontinuity of multiple $\mu$-quasiconvex integrals
ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003), pp. 105-124.

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^{n}}\phantom{\rule{-0.166667em}{0ex}}f\left(x,\phantom{\rule{-0.166667em}{0ex}}{\nabla }_{\mu }u\phantom{\rule{-0.166667em}{0ex}}\right)\mathrm{d}\mu$, where $\mu$ is a given positive measure on ${ℝ}^{n}$, and the vector-valued function $u$ belongs to the Sobolev space ${H}_{\mu }^{1,p}\left({ℝ}^{n},{ℝ}^{m}\right)$ associated with $\mu$. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu$. More precisely, for fully general $\mu$, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu$, turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when $\mu$ belongs to a suitable class of rectifiable measures.

DOI: 10.1051/cocv:2003002
Classification: 28A25,  49J45,  26B25
Keywords: Borel measures, tangent properties, lower semicontinuity
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Fragalà, Ilaria. Lower semicontinuity of multiple $\sf \mu$-quasiconvex integrals. ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003), pp. 105-124. doi : 10.1051/cocv:2003002. http://www.numdam.org/articles/10.1051/cocv:2003002/`

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