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

TL;DR

This paper investigates how fractional gradients in Bessel spaces converge to classical gradients in Sobolev spaces as the fractional parameter approaches one, establishing $ ext{Gamma}$-convergence of related functionals.

Contribution

It provides a rigorous analysis of the convergence of $s$-fractional gradients to classical gradients and proves $ ext{Gamma}$-convergence of polyconvex functionals involving these gradients.

Findings

Strong convergence of $D^s u$ to $Du$ as $s earrow 1

Weak compactness of sequences with bounded $L^p$ norm of $D^s u_s$

Semicontinuity and $ ext{Gamma}$-convergence results for polyconvex functionals

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 $p>n$.