*Published Paper*

**Inserted:** 4 mar 2014

**Last Updated:** 15 may 2016

**Journal:** ESAIM Control Optim. Calc. Var.

**Volume:** 21

**Number:** 4

**Pages:** 1053-1075

**Year:** 2015

**Abstract:**

We study the weak$^*$ lower semicontinuity of functionals of the form \[ F(V)=\mbox{ess sup}_{x \in \Omega} f(x,V (x)) \] where $\Omega\subset \mathbb{R}^N$ is a bounded open set, $V\in L^{\infty}(\Omega;\mathbb{M}^{d\times N})\cap \hbox {Ker} \cal A$ and $\cal A$ is a constant-rank partial differential operator. The notion of $\cal A$-Young quasiconvexity, which is introduced here, provides a sufficient condition when $f(x,\cdot)$ is only lower semicontinuous. We also establish necessary conditions for weak$^*$ lower semicontinuity. Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity.