*Published Paper*

**Inserted:** 24 jun 2000

**Last Updated:** 10 dec 2003

**Journal:** ESAIM: COCV

**Volume:** 9

**Pages:** 105-124

**Year:** 2003

**Abstract:**

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