Lower semicontinuity of multiple $\mu$-quasiconvex integrals

Fragalà, Ilaria

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 105-124.

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where $\mu$ is a given positive measure on ${ℝ}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ 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.

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