标题： 非局部格林定理和截断分数梯度的亥姆霍兹分解 显示英文标题

标题： Nonlocal Green theorems and Helmholtz decompositions for truncated fractional gradients

摘要： 在本工作中，我们进一步发展了一种非局部微积分理论（最初在[5]中引入），该理论与具有有限相互作用支持的核的奇异分数型算子相关。 该框架在非局部弹性理论和无网格动力学理论中的适用性引起了越来越多的关注和研究动力，从而促使人们探索其与经典对应物之间的联系。 特别是，本文的一个关键贡献是推导出向量恒等式、分部积分类型的定理（如散度定理、格林恒等式），以及亥姆霍兹-霍德分解。 这些估计以及在过程中进行的分析为在该框架内证明其他结果提供了基础，并为数值实现提供了途径。 显示英文摘要

摘要： In this work we further develop a nonlocal calculus theory (initially introduced in [5]) associated with singular fractional-type operators which exhibit kernels with finite support of interactions. The applicability of the framework to nonlocal elasticity and the theory of peridynamics has attracted increased interest and motivation to study it and find connections with its classical counterpart. In particular, a critical contribution of this paper is producing vector identities, integration by part type theorems (such as the Divergence Theorem, Green identities), as well as a Helmholtz-Hodge decomposition. The estimates, together with the analysis performed along the way provide stepping stones for proving additional results in the framework, as well as pathways for numerical implementations.