论文标题
在索波列夫空间的新家庭
On New Families of Fractional Sobolev Spaces
论文作者
论文摘要
本文介绍了三个新的Sobolev空间的新家庭及其伴随的理论。新结构和理论基于弱分数衍生物的新概念，这些概念是对整个整数Sobolev空间和理论的自然概括。特别是，引入和分析了两个单方面的Sobolev空间的两个新家族，它们揭示了更多关于另一个所谓的对称分数Sobolev空间家族的见解。在这些SOBOLOLEV空间中，许多关键定理/属性，例如密度/近似定理，扩展定理，单侧跟踪定理以及各种嵌入定理和Sobolev的不等式。此外，还发现了与现有的Sobolev空间的一些关系。本文的结果为系统地开发了变异理论的分数计算和分数PDE理论以及其在后续工作中的数值解决方案奠定了坚实的理论基础。本文是参考文献[7]第1、4和5节的材料的简明介绍。
This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also discovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works. This paper is a concise presentation of the materials of Sections 1, 4 and 5 of reference [7].
