$\Gamma$-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts

Abstract

In this paper we study localization properties of the Riesz $s$-fractional gradient $D^{s}u$ of a vectorial function $u$ as $s\nearrow 1$. The natural space to work with $s$-fractional gradients is the Bessel space $H^{s,p}$ for $0<s<1$ and $1<p<\infty$. This space converges, in a precise sense, to the Sobolev space $W^{1,p}$ when $s\nearrow 1$. We prove that the $s$-fractional gradient $D^{s}u$ of a function $u$ in $W^{1,p}$ converges strongly to the classical gradient $Du$. We also show a weak compactness result in $W^{1,p}$ for sequences of functions $u_s$ with bounded $L^p$ norm of $D^s{u}_s$ as $s\nearrow 1$. Moreover, the weak convergence of $D^s{u}_s$ in $L^p$ implies the weak continuity of its minors, which allows us to prove a semicontinuity result of polyconvex functionals involving $s$-fractional gradients defined in $H^{s,p}$ to their local counterparts defined in $W^{1,p}$. The full $\Gamma$-convergence of the functionals is achieved only for the case $p>n$.