发表机构
University of Nebraska–Lincoln（内布拉斯加大学林肯分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有界域上截断分数阶微分方程的非局部边界条件，证明仅要求函数在领圈上为零会导致分部积分、延拓及不等式等关键性质失效，需引入更小空间，并提出新提升算子、修正格林定理。
AI 中文摘要
非局部模型近年来取得了巨大成功。特别是，用涉及有限相互作用半径的强奇异积分算子替代空间导数的模型，在包括图像处理、连续介质力学以及许多其他领域在内的广泛应用中被使用。然而，从数学分析的角度来看，在考虑合适的“边界”条件概念时会出现困难。（边界的非局部类比通常被称为“领圈”，且通常不是低维集合。）例如，仅仅在领圈上施加齐次狄利克雷型约束，无法保证分析和数值模拟这些模型时所使用的一些基本性质，正如我们在本工作中所展示的那样。在局部情形下，在边界上取零值的函数的索伯列夫空间中工作，允许无边界项地进行分部积分、用测试函数逼近、零延拓到全空间，并应用哈代不等式和庞加莱不等式。我们证明，如果只要求函数在领圈上为零，那么这些性质的每一个非局部类比都会失效，并且必须转移到更小、更具限制性的空间才能享有这些性质。此外，我们引入了一个新的提升算子，纠正了先前发表的关于非局部格林定理的结果中的一个错误，并建立了上述性质与现有分数阶索伯列夫空间概念之间的联系。
英文摘要
Nonlocal models have found great success in recent years. In particular, models that replace spatial derivatives with strongly singular integral operators involving finite interaction radii are used in a wide range of applications, including image processing, continuum mechanics, and many other areas. However, from the perspective of mathematical analysis, difficulties arise when considering the appropriate notion of ``boundary'' conditions. (The nonlocal analogue of the boundary is often called a ``collar'' and is typically not a lower-dimensional set.) For instance, merely enforcing homogeneous Dirichlet-type constraints on the collar fails to guarantee a number of essential properties used in the analysis and numerical simulation of these models, as we show in the present work. In the local setting, working in a Sobolev space of functions that are zero on the boundary allows one to integrate by parts with no boundary term, approximate by test functions, extend by zero to the full space, and apply the Hardy and Poincaré inequalities. We prove that the nonlocal analog of each of these fails if one only requires the functions to be zero on the collar, and that one must move to a smaller, more restrictive space to enjoy these properties. In addition, we introduce a new lifting operator, correct an error in a previously published result on a nonlocal Green's theorem, and establish how the properties listed above relate to existing notions of fractional Sobolev spaces.
Comments36 pages, 2 figures
