Nonlocal Green theorems and Helmholtz decompositions for truncated fractional gradients

2023/11/09 by José C. Bellido, Javier Cueto, Bellido, José C. +5 · 2 citations
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2311.05465

openalex publication_date 2023/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Cited by

• A Lusin theorem for nonlocal gradients
• Local boundary conditions in nonlocal hyperelasticity via heterogeneous horizons

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