论文标题
将线性生长功能扩展到有限分数变化的函数
Extending linear growth functionals to functions of bounded fractional variation
论文作者
论文摘要
在本文中，我们考虑了涉及Riesz分数梯度的新型分数线性生长功能的最小化。这些功能缺乏应用直接方法所需的分数Sobolev空间中的矫正属性。因此，我们利用了最近引入的有界分数变化的空间，并通过相对于弱*收敛性，研究了线性生长功能向这些空间的扩展。我们的主要结果为这种放松建立了明确的表示，其中包括分数变化的单数部分的整体术语，并具有整合体的Quasiconvex信封。准烟气在该部分框架中的作用是通过一种在分数和经典设置之间切换的技术来解释的。我们通过对扩展功能的最小化者的存在理论来补充放松结果。
In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply the direct method. We therefore utilize the recently introduced spaces of bounded fractional variation and study the extension of the linear growth functional to these spaces through relaxation with respect to the weak* convergence. Our main result establishes an explicit representation for this relaxation, which includes an integral term accounting for the singular part of the fractional variation and features the quasiconvex envelope of the integrand. The role of quasiconvexity in this fractional framework is explained by a technique to switch between the fractional and classical settings. We complement the relaxation result with an existence theory for minimizers of the extended functional.
