arXiv 2005.107532020-05-21math.APmath.OC

含s-分数次梯度的多凸泛函的$Γ$ -收敛性

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

José C. Bellido, Javier Cueto, Carlos Mora-Corral

摘要：

本文研究了向量函数$u$的Riesz $s$ -分数阶梯度$D^s u$的局部化性质，即$s \nearrow 1$ .使用$s$ -分数梯度的自然空间是用于$0 < s < 1$和$1 < p < \infty$的贝塞尔空间$H^{s,p}$。这个空间收敛，在一个精确的意义上，索伯列夫空间$W^{1,p}$时，$s \nearrow 1$。证明了函数$u$在$W^{1,p}$中的分数阶梯度$s$-$D^s u$强收敛于经典梯度$Du$ .我们还证明了一个弱紧性结果在$W^{1,p}$的功能序列$u_s$有界$L^p$范数的$D^s u_s$ $s \nearrow 1$。此外，$D^s u_s$在$L^p$中的弱收敛性意味着其子式的弱连续性，这使得我们可以证明包含$H^{s,p}$中定义的$s$ -分数梯度的多凸泛函到$W^{1,p}$中定义的局部对应泛函的连续性.完全$Γ$ -收敛的泛函实现的情况下$p>n$。

英文摘要：

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 $p>n$.

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