$$\Gamma $$-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 \(p>n\).