Calculus of Variations and Geometric Measure Theory
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F. Prinari - N. Ansini

Power-law approximation under differential constraints

created by prinari on 16 Dec 2013
modified by ansini on 11 Nov 2014


Published Paper

Inserted: 16 dec 2013
Last Updated: 11 nov 2014

Volume: 46
Number: 2
Pages: 1085-1115
Year: 2014


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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