Power-Law Approximation under Differential Constraints
Nadia Ansini, Francesca Prinari · SIAM Journal on Mathematical Analysis · 2014
We study the $\Gamma$-convergence, as $p$ tends to $+\infty$, of the power-law functionals $F_p(V)=\left(\int_{\Omega} f^p(x, V(x))dx\right)^{1/p},$ 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 an application of the $\Gamma$-convergence results we characterize the strength set in the context of electrical resistivity.