论文标题
$γ$ - 多凸功能涉及其本地同行的s-fractional梯度的功能
$Γ$-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts
论文作者
论文摘要
在本文中，我们研究了Riesz $ s $ S $级别梯度的本地化属性$ d^s U $ $ u $的$ s \ s \ searl of nearrow 1 $。使用$ S $偏分段梯度的自然空间是$ 0 <s <1 $和$ 1 <p <\ p <\ infty $的贝塞尔空间$ h^{s，p} $。从某种意义上说，这个空间会收敛到$ s \附近1 $时的sobolev space $ w^{1，p} $。我们证明$ s $级梯度$ d^s u $ in函数$ u $ in $ w^{1，p} $强烈收集到经典的梯度$ du $。我们还显示了一个弱的紧凑性结果，$ w^{1，p} $，用于函数序列$ u_s $，带有限制的$ l^p $ norm of $ d^s u_s $ as $ s \ s \ s \ sear nearrow 1 $。此外，$ d^s u_s $在$ l^p $中的弱融合意味着其未成年人的连续性较弱，这使我们能够证明涉及$ h^{s，p} $定义的$ s $ fractional梯度的polyConvex函数的半持续性结果，其本地对应中定义为$ w^^^{1，p}。功能的完整$γ$ - 仅适用于$ p> n $的情况。
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 $Γ$-convergence of the functionals is achieved only for the case $p>n$.
