*Published Paper*

**Inserted:** 16 dec 2013

**Last Updated:** 26 dec 2017

**Journal:** SIAM J MATH ANAL

**Volume:** 46

**Number:** 2

**Pages:** 1085-1115

**Year:** 2014

**Abstract:**

We study the $\Gamma$-convergence of the power-law functionals \[ F_p(V)=\Bigl(\int_{\Omega} f^p(x, V(x))dx\Bigr)^{1/ p}, \] as $p$ tends to $+\infty$, in the setting of constant-rank operator $\cal A$. We show that the $\Gamma$-limit is given by a supremal functional on $L^{\infty}(\Omega;\mathbb{M}^{d\times N}) \cap \hbox {Ker} \cal A$ where $\mathbb{M}^{d\times N}$ is the space of $d\times N$ real matrices. We give an explicit representation formula for the supremand function. We provide some examples and as application of the $\Gamma$-convergence results we characterize the strength set in the context of electrical resistivity.

**Keywords:**
Supremal functionals, $\Gamma$-convergence, $L^p$-approximation, Lower semicontinuity, $\cal A$-quasiconvexity.

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